IC  PROJECTIVE 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


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88CI.-S6006  VINdOdnVO  'S3130NV  SOT 
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Ainiovd  ahvush  ivnoiobu  Nuamnos 

Bjujoiiieo  jo  A}jSjaAjun 


AN  ELEMENTARY  COURSE  IN 

SYNTHETIC  PROJECTIVE 

GEOMETRY 


BY 


DERRICK  NORMAN  LEHMER 

ASSOCIATE   PKOKE.S.SOK  OK  .MATHEMATICS 
I  M\l  IC-HV    OF   CALIKOKMA 


GINX  AND  COMPANY 

BOSTON    •    NEW   YORK    -    CHICAGO    •     LONDON 
ATLANTA     ■     DALLAS     •     COLUMBUS     ■     SAN    KKANi   ISIl) 


COPYRIGHT,  1917,  BY 
DERRICK  NORMAN  LEHMKR 


ALL  RIGHTS  RESERVED 

PRINTED  IN  THE  UNITED  STATES  OF  AMERICA 

226.7 


9%t   fltbtngum   fivtt* 

GINN  AND  COMPANY  •  PRO- 
PRIETORS •  BOSTON  •  U.S.A. 


Engineering  &    /O  J 

statical    W\ 


Sci(  nces 
Library 


-471 


PREFACE 

The  following  course  is  intended  to  give,  in  as  simple 
a  way  as  possible,  the  essentials  of  synthetic  projective 
geometry.  While,  in  the  main,  the  theory  is  developed 
along  the  well-beaten  track  laid  out  by  the  great  masters 
of  the  subject,  it  is  believed  that  there  has  been  a  slight 
smoothing  of  the  road  in  some  places.  Especially  will 
this  be  observed  in  the  chapter  on  Involution.  The 
author  has  never  felt  satisfied  with  the  usual  treatment 
of  that  subject  by  means  of  circles  and  anharmonic 
ratios.  A  purely  projective  notion  ought  not  to  be  based 
on  metrical  foundations.  Metrical  developments  should 
be  made  there,  as  elsewhere  in  the  theory,  by  the 
introduction  of  infinitely  distant  elements. 

The  author  has  departed  from  the  century-old  custom 
of  writing  in  parallel  columns  each  theorem  and  its 
dual.  He  has  not  found  that  it  conduces  to  sharpness 
of  vision  to  try  to  focus  his  eyes  on  two  things  at  once. 
Those  who  prefer  the  usual  method  of  procedure  can, 
of  course,  develop  the  two  sets  of  theorems  side  by  side  ; 
the  author  has  not  found  this  the  better  plan  in  actual 
teaching. 

As  regards  nomenclature,  the  author  has  followed 
the  lead  of  the  earlier  writers  in  English,  and  has  called 
the  system  of  lines  in  a  plane  which  all  pass  through  a 
point  a  pencil  of  rays  instead  of  a  bundle  of  rays,  as  later 
writers   seem    inclined  to  do.     For  a  point  considered 


369803 


iv  PROJECTIVE  GEOMETRY 

as  made  up  of  all  the  lines  and  planes  through  it  he 
has  ventured  to  use  the  term  point  system,  as  being 
the  natural  dualization  of  the  usual  term  j>/<tnr  tyttem. 
He  has  also  rejected  the  terra  foci  of  an  involution,  and 
has  not  used  the  customary  terms  for  classifying  invo- 
lutions —  hyperbolic  involution,  elliptic  involution  and 
parabolic  involution.  He  has  found  that  all  these  terms 
are  very  confusing  to  the  student,  who  inevitably  tries 
to  connect  thera  in  some  way  with  the  conic  sections. 

Enough  examples  have  been  provided  to  give  the 
student  a  clear  grasp  of  the  theory.  Many  are  of  suffi- 
cient generality  to  serve  as  a  basis  for  individual  in- 
vestigation on  the  part  of  the  student.  Thus,  the  third 
example  at  the  end  of  the  first  chapter  will  be  found 
to  be  very  fruitful  in  interesting  results.  A  corre- 
spondence is  there  indicated  between  lines  in  space  and 
circles  through  a  fixed  point  in  space.  If  the  student 
will  trace  a  few  of  the  consequences  of  that  corre- 
spondence, and  determine  what  configurations  of  circles 
correspond  to  intersecting  lines,  to  lines  in  a  plane,  to 
lines  of  a  plane  pencil,  to  lines  cutting  three  skew  lines, 
etc.,  he  will  have  acquired  no  little  practice  in  picturing 
to  himself  figures  in  space. 

The  writer  has  not  followed  the  usual  practice  of 
inserting  historical  notes  at  the  foot  of  the  page,  and 
has  tried  instead,  in  the  last  chapter,  to  give  a  con- 
secutive account  of  the  history  of  pure  geometry,  or,  at 
least,  of  as  much  of  it  as  the  student  will  be  able  to 
appreciate  who  has  mastered  the  course  as  given  in  the 
preceding  chapters.  One  is  not  apt  to  get  a  very  wide 
view  of  the  history  of  a  subject  by  reading  a  hundred 


PREFACE  v 

biographical  footnotes,  arranged  in  no  sort  of  sequence. 
The  writer,  moreover,  feels  that  the  proper  time  to 
learn  the  history  of  a  subject  is  after  the  student  has 
some  general  ideas  of  the  subject  itself. 

The  course  is  not  intended  to  furnish  an  illustration 
of  how  a  subject  may  be  developed  from  the  smallest 
possible  number  of  fundamental  assumptions.  The 
author  is  aware  of  the  importance  of  work  of  this  sort, 
but  he  does  not  believe  it  is  possible  at  the  present 
time  to  write  a  book  along  such  lines  which  shall  be  of 
much  use  for  elementary  students.  For  the  purposes  of 
this  course  the  student  should  have  a  thorough  ground- 
ing in  ordinary  elementary  geometry  so  far  as  to  include 
the  study  of  the  circle  and  of  similar  triangles.  No  solid 
geometry  is  needed  beyond  the  little  used  in  the  proof 
of  Desargues'  theorem  (25),  and,  except  in  certain 
metrical  developments  of  the  general  theory,  there  will 
be  no  call  for  a  knowledge  of  trigonometry  or  analytical 
geometry.  Naturally  the  student  who  is  equipped  with 
these  subjects  as  well  as  with  the  calculus  will  be  a 
little  more  mature,  and  may  be  expected  to  follow  the 
course  all  the  more  easily.  The  author  has  had  no 
difficulty,  however,  in  presenting  it  to  students  in  the 
freshman  class  at  the  University  of  California. 

The  subject  of  synthetic  projective  geometry  is,  in 
the  opinion  of  the  writer,  destined  shortly  to  force  its 
way  down  into  the  secondaiy  schools ;  and  if  this  little 
book  helps  to  accelerate  the  movement,  he  will  feel 
amply  repaid  for  the  task  of  working  the  materials  into 
a  form  available  for  such  schools  as  well  as  for  the 
lower  classes  in  the  university. 


vi  PROJECTIVE  GEOMETRY 

The  material  for  the  course  has  been  drawn  from 
many  sources.  The  author  is  chiefly  indebted  to  the 
classical  works  of  Reye,  Cremona,  Steiner,  Poncelet,  and 
Von  Staudt.  Acknowledgments  and  thanks  are  also 
due  to  Professor  Walter  C.  Eells,  of  the  U.S.  Naval 
Academy  at  Annapolis,  for  his  searching  examination 
and  keen  criticism  of  the  manuscript ;  also  to  Professor 
Herbert  Ellsworth  Slaught,  of  The  University  of  Chicago, 
for  his  many  valuable  suggestions,  and  to  Professor 
B.  M.  Woods  and  Dr.  H.  N.  Wright,  of  the  University 
of  California,  who  have  tried  out  the  methods  of 
presentation  in  their  own  classes. 

D.  X.   LKIIMER 
Bkkkelky,  California 


CONTENTS 


CHAPTER  I 
ONE-TO-ONE  CORRESPONDENCE 

SECTION  PAGE 

1.  Definition  of  one-to-one  correspondence 1 

2.  Consequences  of  one-to-one  correspondence 2 

3.  Applications  in  mathematics 2 

4.  One-to-one  correspondence  and  enumeration 3 

5.  Correspondence  between  a  part  and  the  whole       4 

6.  Infinitely  distant  point 4 

7.  Axial  pencil  ;  fundamental  forms 6 

8.  Perspective  position 5 

0.    Projective  relation 6 

10.  Infinity-to-one  correspondence 7 

11.  Infinitudes  of  different  orders 7 

12.  Points  in  a  plane 8 

13.  Lines  through  a  point 8 

14.  Planes  through  a  point 8 

15.  Lines  in  a  plane 8 

16.  Plane  system  and  point  system 9 

17.  Planes  in  space 9 

18.  Points  in  space 9 

19.  Space  system 9 

20.  Lines  in  space 10 

21.  Correspondence  between  points  and  numbers 10 

22.  Elements  at  infinity 11 

Problems 12 

CHAPTER  IT 

RELATIONS  BETWEEN  FUNDAMENTAL  FORMS  IN  ONE- 
TO-ONE  CORRESPONDENCE  WITH  EACH  OTHER 

23.  Seven  fundamental  forms 14 

24.  Projective  properties 14 

25.  Desargues's  theorem 15 

vii 


viii  PROJECTIVE  GEOMETRY 

8ECTION  PAGE 

26.  Fundamental  theorem  concerning  two  complete  quadrangles  16 

27.  Importance  of  the  theorem 17 

28.  Restatement  of  the  theorem 18 

29.  Four  harmonic  points 18 

30.  Harmonic  conjugates 19 

31.  Importance  of  the  notion  of  four  harmonic  points     ....  19 

32.  Projective  invariance  of  four  harmonic  points 20 

33.  Four  harmonic  lines 20 

34.  Four  harmonic  planes 20 

35.  Summary  of  results 21 

36.  Definition  of  projectivity 21 

37.  Correspondence  between  harmonic  conjugates 21 

38.  Separation  of  harmonic  conjugates 22 

39.  Harmonic  conjugate  of  the  point  at  infinity 23 

40.  Projective  theorems  and  metrical   theorems.     Linear  con- 

struction      23 

41.  Parallels  and  mid-points 24 

42.  Division  of  a  segment  into  equal  parts 25 

43.  Numerical  relations 25 

44.  Algebraic  formula  connecting  four  harmonic  points  ....  25 

45.  Further  formula? 26 

46.  Anharmonic  ratio 27 

Problems 27 


CHAPTER  III 

COMBINATION  OF  TWO  PROJECTIVELY  RELATED 
FUNDAMENTAL  FORMS 

47.  Superposed  fundamental  forms.  Self-corresponding  elements  29 

48.  Special  case 30 

49.  Fundamental  theorem.   Postulate  of  continuity 31 

50.  Extension  of  theorem  to  pencils  of  rays  and  planes   ....  32 

51.  Projective  point-rows  having  a  self-corresponding  point    .     .  32 

52.  Point-rows  in  perspective  position 33 

53.  Pencils  in  perspective  position 33 

54.  Axial  pencils  in  perspective  position 33 

55.  Point-row  of  the  second  order 33 

56.  Degeneration  of  locus 34 


CONTENTS  ix 

SECTION  PAGE 

57.  Pencils  of  rays  of  the  second  order 34 

58.  Degenerate  case 34 

69.   Cone  of  the  second  order 35 

Problems 35 

CHAPTER  IV 
POINT-ROWS  OF  THE  SECOND  ORDER 

60.  Point-row  of  the  second  order  defined 37 

61.  Tangent  line 37 

62.  Determination  of  the  locus 38 

63.  Restatement  of  the  problem 38 

64.  Solution  of  the  fundamental  problem 38 

65.  Different  constructions  for  the  figure 39 

66.  Lines  joining  four  points  of  the  locus  to  a  fifth 40 

67.  Restatement  of  the  theorem 40 

68.  Further  important  theorem 40 

69.  Pascal's  theorem 40 

70.  Permutation  of  points  in  Pascal's  theorem 41 

71.  Harmonic  points  on  a  point-row  of  the  second  order      ...  42 

72.  Determination  of  the  locus 42 

73.  Cii'cles  and  conies  as  point-rows  of  the  second  order      ...  43 

74.  Conic  through  five  points 43 

75.  Tangent  to  a  conic 44 

76.  Inscribed  quadrangle 44 

77.  Inscribed  triangle 45 

78.  Degenerate  conic 46 

Problems 46 

CHAPTER  V 
PENCILS  OF  RAYS  OF  THE  SECOND  ORDER 

79.  Pencil  of  rays  of  the  second  order  defined 48 

80.  Tangents  to  a  circle 48 

81.  Tangents  to  a  conic 49 

82.  Generating  point-rows  lines  of  the  system 49 

83.  Determination  of  the  pencil 49 

84.  Brianchon's  theorem 51 


x  PROJECTIVE  GEOMETRY 

SECTION  PAGE 

85.  Permutation  of  lines  in  Brianchon's  theorem 51 

86.  Construction  of  the  pencil  by  Brianchon's  theorem  ....  51 

87.  Point  of  contact  of  a  tangent  to  a  conic 52 

88.  Circumscribed  quadrilateral 52 

89.  Circumscribed  triangle 53 

90.  Use  of  Brianchon's  theorem 53 

91.  Harmonic  tangents 53 

92.  Projectivity  and  perspectivity 53 

93.  Degenerate  case 54 

94.  Law  of  duality 54 

Problems 64 


CHAPTER  VI 
POLES  AND  POLARS 

95.  Inscribed  and  circumscribed  quadrilaterals 56 

96.  Definition  of  the  polar  line  of  a  point 56 

97.  Further  defining  properties 57 

98.  Definition  of  the  pole  of  a  line 57 

99.  Fundamental  theorem  of  poles  and  polars 57 

100.  Conjugate  points  and  lines 57 

101.  Construction  of  the  polar  line  of  a  given  point 58 

102.  Self-polar  triangle 58 

103.  Pole  and  polar  projectively  related 58 

104.  Duality 59 

105.  Self-dual  theorems 60 

106.  Other  correspondences 60 

Problems 60 


CHAPTER  VII 
METRICAL  PROPERTIES  OF  THE  CONIC  SECTIONS 

107.  Diameters.   Center 62 

108.  Various  theorems 62 

109.  Conjugate  diameters 62 

110.  Classification  of  conies 63 

111.  Asymptotes 63 


CONTENTS  xi 

SECTION  PAijE 

112.  Various  theorems 63 

113.  Theorems  concerning  asymptotes 63 

114.  Asymptotes  and  conjugate  diameters 64 

115.  Segments  cut  off  on  a  chord  by  hyperbola  and  its  asymp- 

totes        64 

116.  Application  of  the  theorem 64 

117.  Triangle  formed  by  the  two  asymptotes  and  a  tangent    .    .  65 

118.  Equation  of  hyperbola  referred  to  the  asymptotes  ....  65 

119.  Equation  of  parabola 66 

120.  Equation  of  central  conies  referred  to  conjugate  diameters  68 
Problems 70 


CHAPTER  VIII 
INVOLUTION 

121.  Fundamental  theorem 71 

122.  Linear  construction 72 

123.  Definition  of  involution  of  points  on  a  line 72 

124.  Double-points  in  an  involution 73 

125.  Desargues's  theorem  concerning  conies  through  four  points  .  74 

126.  Degenerate  conies  of  the  system 74 

127.  Conies  through  four  points  touching  a  given  line      ....  75 

128.  Double  correspondence 75 

129.  Steiner's  construction 76 

130.  Application  of  Steiner's  construction  to  double  correspond- 

ence    77 

131.  Involution  of  points  on  a  point-row  of  the  second  order       .  78 

132.  Involution  of  rays 79 

133.  Double  rays 80 

134.  Conic  through  a  fixed  point  touching  four  lines 80 

135.  Double  correspondence 80 

136.  Pencils  of  rays  of  the  second  order  in  involution      ....  81 

137.  Theorem  concerning  pencils  of  the  second  order  in  involu- 

tion    81 

138.  Involution  of  rays  determined  by  a  conic 81 

139.  Statement  of  theorem 81 

140.  Dual  of  the  theorem 82 

Problems 82 


xii  PROJECTIVE  GEOMETRY 

CHAPTER  IX 
METRICAL  PROPERTIES  OF  INVOLUTIONS 

SECTION  PAGE 

141.  Introduction  of  infinite  point ;   center  of  involution  ...  84 

142.  Fundamental  metrical  theorem 85 

143.  Existence  of  double  points 85 

144.  Existence  of  double  rays 86 

145.  Construction  of  an  involution  by  means  of  circles  ....  86 

146.  Circular  points 87 

147.  Pairs  in  an  involution  of  rays  which  are  at  right  angles. 

Circular  involution 88 

148.  Axes  of  conies 88 

149.  Points  at  which  the  involution  of   rays  determined  by  a 

conic  is  circular 89 

150.  Properties  of  such  a  point 90 

151.  Position  of  such  a  point 90 

152.  Discovery  of  the  foci  of  the  conic 91 

153.  The  circle  and  the  parabola 92 

154.  Focal  properties  of  conies 93 

155.  Case  of  the  parabola 94 

156.  Parabolic  reflector 94 

157.  Directrix.    Principal  axis.    Vertex 94 

158.  Another  definition  of  a  conic 94 

159.  Eccentricity 95 

160.  Sum  or  difference  of  focal  distances 95 

Problems 96 

CHAPTER  X 
ON  THE  HISTORY  OF  SYNTHETIC  PROJECTIVE  GEOMETRY 

161.  Ancient  results      98 

162.  Unifying  principles 101 

163.  Desargues 101 

164.  Poles  and  polars 102 

165.  Desargues's  theorem  concerning  conies  through  four  points  102 

166.  Extension  of  the  theory  of  poles  and  polars  to  space      .    .  103 

167.  Desargues's  method  of  describing  a  conic 104 

168.  Reception  of  Desargues's  work 104 


CONTENTS  xiii 

SECTION  PAGE 

169.  Conservatism  in  Desargues's  time 105 

170.  Desargues's  style  of  writing 105 

171.  Lack  of  appreciation  of  Desargues 107 

172.  Pascal  and  his  theorem 108 

173.  Pascal's  essay 108 

174.  Pascal's  originality 109 

175.  De  la  Hire  and  his  work 109 

176.  Descartes  and  his  influence Ill 

177.  Newton  and  Maclaurin 112 

178.  Maclaurin's  construction      112 

179.  Descriptive  geometry  and  the  second  revival 113 

180.  Duality,  homology,  continuity,  contingent  relations  ...  114 

181.  Poncelet  and  Cauchy 115 

182.  The  work  of  Poncelet 116 

183.  The    debt   which    analytic   geometry   owes   to    synthetic 

geometry 116 

184.  Steiner  and  his  work 117 

185.  Von  Staudt  and  his  work 118 

186.  Recent  developments 119 

INDEX 121 


AN  ELEMENTARY  COURSE  IN 

SYNTHETIC  PROJECTIVE 

GEOMETRY 

CHAPTER   I 

ONE-TO-ONE  CORRESPONDENCE 

1.  Definition  of  one-to-one  correspondence.  Given  any 
two  sets  of  individuals,  if  it  is  possible  to  set  up  such 
a  correspondence  between  the  two  sets  that  to  any 
individual  in  one  set  corresponds  one  and  only  one 
individual  in  the  other,  then  the  two  sets  are  said  to 
be  in  one-to-one  correspondence  with  each  other.  This 
notion,  simple  as  it  is,  is  of  fundamental  importance 
in  all  branches  of  science.  The  process  of  counting  is 
nothing  but  a  setting  up  of  a  one-to-one  correspond- 
ence between  the  objects  to  be  counted  and  certain 
words,  '  one,'  r  two,'  '  three,'  etc.,  in  the  mind.  Many 
savage  peoples  have  discovered  no  better  method  of 
counting  than  by  setting  up  a  one-to-one  correspondence 
between  the  objects  to  be  counted  and  their  fingers. 
The  scientist  who  busies  himself  with  naming  and 
classifying  the  objects  of  nature  is  only  set-ting  up  a 
one-to-one  correspondence  between  the  objects  and  cer- 
tain words  which  serve,  not  as  a  means  of  counting  the 

l 


2  PROJECTIVE  GEOMETRY 

objects,  but  of  listing  them  in  a  convenient  way.  Thus 
he  may  be  able  to  marshal  and  array  his  material  in 
such  a  way  as  to  bring  to  light  relations  that  may 
exist  between  the  objects  themselves.  Indeed,  the  whole 
notion  of  language  springs  from  this  idea  of  one-to-one 
correspondence. 

2.  Consequences  of  one-to-one  correspondence.  The 
most  useful  and  interesting  problem  that  may  arise  in 
connection  with  any  one-to-one  correspondence  is  to 
determine  just  what  relations  existing  between  the 
individuals  of  one  assemblage  may  be  carried  over  to 
another  assemblage  in  one-to-one  correspondence  with 
it.  It  is  a  favorite  error  to  assume  that  whatever  holds 
for  one  set  must  also  hold  for  the  other.  Magicians  are 
apt  to  assign  magic  properties  to  many  of  the  words 
and  symbols  which  they  are  in  the  habit  of  using,  and 
scientists  are  constantly  confusing  objective  things  with 
the  subjective  formulas  for  them.  After  the  physicist 
has  set  up  correspondences  between  physical  facts  and 
mathematical  formulas,  the  M  interpretation "  of  these 
formulas  is  his  most  important  and  difficult  task. 

3.  In  mathematics,  effort  is  constantly  being  made 
to  set  up  one-to-one  correspondences  between  simple 
notions  and  more  complicated  ones,  or  between  the  well- 
explored  fields  of  research  and  fields  less  known.  Thus, 
by  means  of  the  mechanism  employed  in  analytic  geom- 
etry, algebraic  theorems  are  made  to  yield  geometric 
ones,  and  vice  versa.  In  geometry  we  get  at  the  proper- 
ties of  the  conic  sections  by  means  of  the  properties 
of  the  straight  line,  and  cubic  surfaces  are  studied  by 
means  of  the  plane. 


ONE-TO-ONE  CORRESPONDENCE 


4.  One-to-one  correspondence  and  enumeration.  If  a 
one-to-one  correspondence  has  been  set  up  between  the 
objects  of  one  set  and  the  objects  of  another  set,  then 
the  inference  may  usually  be  drawn  that  they  have  the 
same  number  of  elements.  If,  however,  there  is  an 
infinite  number  of  individuals  in 
each  of  the  two  sets,  the  notion 
of  counting  is  necessarily  ruled 
out.  It  may  be  possible,  never- 
theless, to  set  up  a  one-to-one 
correspondence  between  the  ele- 
ments of  two  sets  even  when  the 
number  is  infinite.  Thus,  it  is  easy  to  set  up  such  a 
correspondence  between  the  points  of  a  line  an  inch 
long  and  the  points  of  a  line  two  inches  long.  For  let 
the  lines  (Fig.  1)  be  AB  and  A'B'.  Join  AA'  and  BB', 
and  let  these  joining  lines  meet  in  >$'.  For  every  point  C 
on  AB  a  point  C'  may  be  found 
on  A'B'  by  joining  C  to  S  and 
noting  the  point  C  where  CS 
meets  A'B'.  Similarly,  a  point  C 
may  be  found  on  AB  for  any 
point  C  on  A'B'.  The  corre- 
spondence is  clearly  one-to-one, 
but  it  would  be  absurd  to  infer 
from  this  that  there  were  just 

as  man}*  points  on  AB  as  on  A'B'.  In  fact,  it  would 
be  just  as  reasonable  to  infer  that  there  were  twice  as 
many  points  on  A'B'  as  on  AB.  For  if  we  bend  A'B' 
into  a  circle  with  center  at  S  (Fig.  2),  we  see  that  for 
every  point  C  on  AB  there  are  two  points  on  A'B'.   Thus 


4  PROJECTIVE  GEOMETRY 

it  is  seen  that  the  notion  of  one-to-one  correspondence 
is  more  extensive  than  the  notion  of  counting,  and 
includes  the  notion  of  counting  only  when  applied  to 
finite  assemblages. 

5.  Correspondence  between  a  part  and  the  whole  of  an 
infinite  assemblage.  In  the  discussion  of  the  last  para- 
graph the  remarkable  fact  was  brought  to  light  that  it 
is  sometimes  possible  to  set  the  elements  of  an  assem- 
blage into  one-to-one  correspondence  with  a  part  of 
those  elements.  A  moment's  reflection  will  convince 
one  that  this  is  never  possible  when  there  is  a  finite 
number  of  elements  in  the  assemblage.  Indeed,  we 
may  take  this  property  as  our  definition  of  an  infinite 
assemblage,  and  say  that  an  infinite  assemblage  is  one 
that  may  be  put  into  one-to-one  correspondence  with 
part  of  itself.  This  has  the  advantage  of  being  a  positive 
definition,  as  opposed  to  the  usual  negative  definition  of 
an  infinite  assemblage  as  one  that  cannot  be  counted. 

6.  Infinitely  distant  point.  We  have  illustrated  above 
a  simple  method  of  setting  the  points  of  two  lines  into 
one-to-one  correspondence.  The  same  illustration  will 
serve  also  to  show  how  it  is  possible  to  set  the  points 
on  a  line  into  one-to-one  correspondence  with  the  lines 
through  a  point.  Thus,  for  any  point  C  on  the  line  AB 
there  is  a  line  SC  through  S.  We  must  assume  the  line 
AB  extended  indefinitely  in  both  directions,  however,  if 
we  are  to  have  a  point  on  it  for  every  line  through  S; 
and  even  with  this  extension  there  is  one  line  through 
S,  according  to  Euclid's  postulate,  which  does  not  meet 
the  line  AB  and  which  therefore  has  no  point  on 
AB  to  correspond  to  it.     In  order  to  smooth  out  this 


ONE-TO-ONE  CORRESPONDENCE  5 

discrepancy  we  are  accustomed  to  assume  the  existence 
of  an  infinitely  distant  point  on  the  line  AB  and  to  assign 
this  point  as  the  corresponding  point  of  the  exceptional 
line  of  S.  With  this  understanding,  then,  we  may  say 
that  we  have  set  the  lines  through  a  point  and  the 
points  on  a  line  into  one-to-one  correspondence.  This 
correspondence  is  of  such  fundamental  importance  in 
the  study  of  projective  geometry  that  a  special  name  is 
given  to  it.  Calling  the  totality  of  points  on  a  line  a 
point-roiv,  and  the  totality  of  lines  through  a  point  a 
pencil  of  rays,  we  say  that  the  point-row  and  the  pencil 
related  as  above  are  hi  perspective  position,  or  that  they 
are  perspectively  related. 

7.  Axial  pencil ;  fundamental  forms.  A  similar  cor- 
respondence may  be  set  up  between  the  points  on  a 
line  and  the  planes  through  another  line  which  does  not 
meet  the  first.  Such  a  system  of  planes  is  called  an 
axial  pencil,  and  the  three  assemblages  —  the  point-row, 
the  pencil  of  rays,  and  the  axial  pencil — are  called 
fundamental  forms.  The  fact  that  they  may  all  be  set 
into  one-to-one  correspondence  with  each  other  is  ex- 
pressed by  saying  that  they  are  of  the  same  order.  It  is 
usual  also  to  speak  of  them  as  of  the  first  order.  We 
shall  see  presently  that  there  are  other  assemblages 
which  cannot  be  put  into  this  sort  of  one-to-one  cor- 
respondence with  the  points  on  a  line,  and  that  they 
will  very  reasonably  be  said  to  be  of  a  higher  order. 

8.  Perspective  position.  We  have  said  that  a  point- 
row  and  a  pencil  of  ra}-s  are  in  perspective  position  if 
each  ray  of  the  pencil  goes  through  the  point  of  the 
point-row  which  corresponds  to  it.    Two  pencils  of  rays 


6  PROJECTIVE  GEOMETRY 

are  also  said  to  be  in  perspective  position  if  correspond- 
ing rays  meet  on  a  straight  line  which  is  called  the 
axis  of  perspectivity.  Also,  two  point-rows  are  said  to 
be  in  perspective  position  if  corresponding  points  lie  on 
straight  lines  through  a  point  which  is  called  the  center 
of  perspectivity.  A  point-row  and  an  axial  pencil  are 
in  perspective  position  if  each  plane  of  the  pencil  goes 
through  the  point  on  the  point-row  which  corresponds 
to  it,  and  an  axial  pencil  and  a  pencil  of  rays  are  in 
perspective  position  if  each  ray  lies  in  the  plane  which 
corresponds  to  it;  and,  finally,  two  axial  pencils  are 
perspectively  related  if  corresponding  planes  meet  in 
a  plane. 

9.  Projective  relation.  It  is  easy  to  imagine  a  more 
general  correspondence  between  the  points  of  two  point- 
rows  than  the  one  just  described.  If  we  take  two 
perspective  pencils,  A  and  B,  then  a  point-row  a  per- 
spective to  A  will  be  in  one-to-one  correspondence  with 
a  point-row  b  perspective  to  2?,  but  corresponding  points 
will  not,  in  general,  lie  on  lines  which  all  pass  through 
a  point.  Two  such  point-rows  are  said  to  be  projectively 
related,  or  simply  projective  to  each  other.  Similarly, 
two  pencils  of  rays,  or  of  planes,  are  projectively  related 
to  each  other  if  they  are  perspective  to  two  perspective 
point-rows.  This  idea  will  be  generalized  later  on.  It  is 
important  to  note  that  between  the  elements  of  two 
projective  fundamental  forms  there  is  a  one-to-one  cor- 
respondence, and  also  that  this  correspondence  is  in 
general  continuous;  that  is,  by  taking  two  elements  of 
one  form  sufficiently  close  to  each  other,  the  two  corre- 
sponding elements  in  the  other  form  may  be  made  to 


ONE-TO-ONE  CORRESPONDENCE  7 

approach  each  other  arbitrarily  close.  In  the  case  of 
point-rows  this  continuity  is  subject  to  exception  in  the 
neighborhood  of  the  point  "at  infinity." 

10.  Infinity-to-one  correspondence.  It  might  be  inferred 
that  any  infinite  assemblage  could  be  put  into  one-to-one 
correspondence  with  any  other.  Such  is  not  the  case, 
however,  if  the  correspondence  is  to  be  continuous, 
between  the  points  on  a  line  and  the  points  on  a  plane. 
Consider  two  lines  which  lie  in  different  planes,  and 
take  m  points  on  one  and  n  points  on  the  other.  The 
number  of  lines  joining  the  m  points  of  one  to  the 
n  points  of  the  other  is  clearly  mn.  If  we  symbolize 
the  totality  of  points  on  a  line  by  oo,  then  a  reasonable 
symbol  for  the  totality  of  lines  drawn  to  cut  two  lines 
would  be  co2.  Clearly,  for  every  point  on  one  line  there 
are  oo  lines  cutting  across  the  other,  so  that  the  corre- 
spondence might  be  called  oo-to-one.  Thus  the  assem- 
blage of  lines  cutting  across  two  lines  is  of  higher 
order  than  the  assemblage  of  points  on  a  line ;  and  as 
we  have  called  the  point-row  an  assemblage  of  the  first 
order,  the  system  of  lines  cutting  across  two  lines  ought 
to  be  called  of  the  second  order. 

11.  Infinitudes  of  different  orders.  Now  it  is  easy  to 
set  up  a  one-to-one  correspondence  between  the  points 
in  a  plane  and  the  system  of  lines  cutting  across  two 
lines  which  lie  in  different  planes.  In  fact,  each  line  of 
the  system  of  lines  meets  the  plane  in  one  point,  and 
each  point  in  the  plane  determines  one  and  only  one  line 
cutting  across  the  two  given  lines  —  namely,  the  line  of 
intersection  of  the  two  planes  determined  by  the  given 
point  with   each    of  the  given   lines.    The   assemblage 


8  PROJECTIVE  GEOMETRY 

of  points  in  the  plane  is  thus  of  the  same  order  as 
that  of  the  lines  cutting  across  two  lines  which  lie  in 
different  planes,  and  ought  therefore  to  be  spoken  of 
as  of  the  second  order.  We  express  all  these  results 
as  follows: 

12.  If  the  infinitude  of  points  on  a  line  is  taken  as 
the  infinitude  of  the  first  order,  then  the  infinitude  of 
lines  in  a  pencil  of  rays  and  the  infinitude  of  planes  in 
an  axial  pencil  are  also  of  the  first  order,  while  the 
infinitude  of  lines  cutting  across  two  "skew"  lines,  as 
well  as  the  infinitude  of  points  in  a  plane,  are  of  the 
second  order. 

13.  If  we  join  each  of  the  points  of  a  plane  to  a  point 
not  in  that  plane,  we  set  up  a  one-to-one  correspondence 
between  the  points  in  a  plane  and  the  lines  through 
a  point  in  space.  Thus  the  infinitude  of  lines  through  a 
point  in  space  is  of  the  second  order. 

14.  If  to  each  line  through  a  point  in  space  we  make 
correspond  that  plane  at  right  angles  to  it  and  passing 
through  the  same  point,  we  see  that  the  infinitude  of 
planes  through  a  point  in  space  is  of  the  second  order. 

15.  If  to  each  plane  through  a  point  in  space  we 
make  correspond  the  line  in  which  it  intersects  a  given 
plane,  we  see  that  the  infinitude  of  lines  in  a  plane  is  of 
the  second  order.  This  may  also  be  seen  by  setting  up 
a  one-to-one  correspondence  between  the  points  on  a 
plane  and  the  lines  of  that  plane.  Thus,  take  a  point  S 
not  in  the  plane.  Join  any  point  M  of  the  plane  to  S. 
Through  S  draw  a  plane  at  right  angles  to  MS.  This 
meets  the  given  plane  in  a  line  m  which  may  be  taken  as 
corresponding  to  the  point  M.    Another  very  important 


ONE-TO-ONE  CORRESPONDENCE  9 

method  of  setting  up  a  one-to-one  correspondence  be- 
tween lines  and  points  in  a  plane  will  be  given  later,  and 
many  weighty  consequences  will  be  derived  from  it. 

16.  Plane  system  and  point  system.  The  plane,  con- 
sidered as  made  up  of  the  points  and  lines  in  it,  is  called 
a  plane  system  and  is  a  fundamental  form  of  the  second 
order.  The  point,  considered  as  made  up  of  all  the  lines 
and  planes  passing  through  it,  is  called  a  point  system 
and  is  also  a  fundamental  form  of  the  second  order. 

17.  If  now  we  take  three  lines  in  space  all  lying  in 
different  planes,  and  select  I  points  on  the  first,  m  points 
on  the  second,  and  n  points  on  the  third,  then  the  total 
number  of  planes  passing  through  one  of  the  selected 
points  on  each  line  will  be  Imn.  It  is  reasonable,  there- 
fore, to  symbolize  the  totality  of  planes  that  are  deter- 
mined by  the  go  points  on  each  of  the  three  lines  by 
go3,  and  to  call  it  an  infinitude  of  the  third  order.  But 
it  is  easily  seen  that  every  plane  in  space  is  included  in 
this  totality,  so  that  the  totality  of  planes  in  space  is  an 
infinitude  of  the  third  order. 

18.  Consider  now  the  planes  perpendicular  to  these 
three  lines.  Every  set  of  three  planes  so  drawn  will 
determine  a  point  in  space,  and,  conversely,  through 
every  point  in  space  may  be  drawn  one  and  only  one 
set  of  three  planes  at  right  angles  to  the  three  given 
lines.  It  follows,  therefore,  that  the  totality  of  points 
in  space  is  an  infinitude  of  the  third  order. 

19.  Space  system.  Space  of  three  dimensions,  con- 
sidered as  made  up  of  all  its  planes  and  points,  is  then 
a  fundamental  form  of  the  third  order,  which  we  shall 
call  a  space  system. 


10  PROJECTIVE  GEOMETRY 

20.  Lines  in  space.  If  we  join  the  twofold  infinity 
of  points  in  one  plane  with  the  twofold  infinity  of 
points  in  another  plane,  we  get  a  totality  of  lines  of 
space  which  is  of  the  fourth  order  of  infinity.  The 
totality  of  lines  in  space  gives,  then,  a  fundamental  form 
of  the  fourth  order. 

21.  Correspondence  between  points  and  numbers.  In 
the  theory  of  analytic  geometry  a  one-to-one  corre- 
spondence is  assumed  to  exist  between  points  on  a 
line  and  numbers.  In  order  to  justify  this  assumption 
a  very  extended  definition  of  number  must  be  made 
use  of.  A  one-to-one  correspondence  is  then  set  up  be- 
tween points  in  the  plane  and  pairs  of  numbers,  and 
also  between  points  in  space  and  sets  of  three  numbers. 
A  single  constant  will  serve  to  define  the  position  of 
a  point  on  a  line ;  two,  a  point  in  the  plane ;  three,  a 
point  in  space ;  etc.  In  the  same  theory  a  one-to-one 
correspondence  is  set  up  between  loci  in  the  plane  and 
equations  in  two  variables ;  between  surfaces  in  space 
and  equations  in  three  variables ;  etc.  The  equation  of 
a  line  in  a  plane  involves  two  constants,  either  of  which 
may  take  an  infinite  number  of  values.  From  this  it 
follows  that  there  is  an  infinity  of  lines  in  the  plane 
which  is  of  the  second  order  if  the  infinity  of  points  on 
a  line  is  assumed  to  be  of  the  first.  In  the  same  way 
a  circle  is  determined  by  three  conditions ;  a  sphere  by 
four ;  etc.  We  might  then  expect  to  be  able  to  set  Dp 
a  one-to-one  correspondence  between  circles  in  a  plane 
and  points,  or  planes  in  space,  or  between  spheres  and 
lines  in  space.  Such,  indeed,  is  the  case,  and  it  is 
often    possible    to    infer    theorems    concerning   spheres 


ONE-TO-ONE  CORRESPONDENCE  11 

from  theorems  concerning  lines,  and  vice  versa.  It  is 
possibilities  such  as  these  that  give  to  the  theory  of 
one-to-one  correspondence  its  great  importance  for  the 
mathematician.  It  must  not  be  forgotten,  however,  that 
we  are  considering  only  continuous  correspondences.  It 
is  perfectly  possible  to  set  up  a  one-to-one  correspond- 
ence between  the  points  of  a  line  and  the  points  of  a 
plane,  or,  indeed,  between  the  points  of  a  line  and  the 
points  of  a  space  of  any  finite  number  of  dimensions,  if 
the  correspondence  is  not  restricted  to  be  continuous. 

22.  Elements  at  infinity.  A  final  word  is  necessary 
in  order  to  explain  a  phrase  which  is  in  constant  use  in 
the  study  of  projective  geometry.  We  have  spoken  of 
the  "  point  at  infinity "  on  a  straight  line  —  a  fictitious 
point  only  used  to  bridge  over  the  exceptional  case 
when  we  are  setting  up  a  one-to-one  correspondence 
between  the  points  of  a  line  and  the  lines  through  a 
point.  We  speak  of  it  as  "a  point"  and  not  as  "points," 
because  in  the  geometry  studied  by  Euclid  we  assume 
only  one  line  through  a  point  parallel  to  a  given  line. 
In  the  same  sense  we  speak  of  all  the  points  at  infinity 
in  a  plane  as  lying  on  a  line,  "the  line  at  infinity," 
because  the  straight  line  is  the  simplest  locus  we  can 
imagine  which  has  only  one  point  in  common  with  any 
line  in  the  plane.  Likewise  we  speak  of  the  "  plane  at 
infinity,"  because  that  seems  the  most  convenient  way 
of  imagining  the  points  at  infinity  in  space.  It  must  not 
be  inferred  that  these  conceptions  have  any  essential 
connection  with  physical  facts,  or  that  other  means  of 
picturing  to  ourselves  the  infinitely  distant  configura- 
tions are  not  possible.  In  other  branches  of  mathematics, 


12  PROJECTIVE  GEOMETRY 

notably  in  the  theory  of  functions  of  a  complex  vari- 
able, quite  different  assumptions  are  made  and  quite 
different  conceptions  of  the  elements  at  infinity  are  used. 
As  we  can  know  nothing  experimentally  about  such 
things,  we  are  at  liberty  to  make  any  assumptions  we 
please,  so  long  as  they  are  consistent  and  serve  some 
useful  purpose. 

PROBLEMS 

1.  Since  there  is  a  threefold  infinity  of  points  in  space, 
there  must  be  a  sixfold  infinity  of  pairs  of  points  in  space. 
Each  pair  of  points  determines  a  line.  Why,  then,  is  there 
not  a  sixfold  infinity  of  lines  in  space  ? 

2.  If  there  is  a  fourfold  infinity  of  lines  in  Bpace,  why 
is  it  that  there  is  not  a  fourfold  infinity  of  planes  through 
a  point,  seeing  that  each  line  in  space  determines  a  plane 
through  that  point  ? 

3.  Show  that  there  is  a  fourfold  infinity  of  circles  in 
space  that  pass  through  a  fixed  point.  (Set  up  a  one-to-one 
correspondence  between  the  axes  of  the  circles  and  lines 
in  space.) 

4.  Find  the  order  of  infinity  of  all  the  lines  of  space 
that  cut  across  a  given  line ;  across  two  given  lines  ;  across 
three  given  lines ;  across  four  given  lines. 

5.  Find  the  order  of  infinity  of  all  the  spheres  in  space 
that  pass  through  a  given  point ;  through  two  given  points  ; 
through  three  given  points  ;  through  four  given  points. 

6.  Find  the  order  of  infinity  of  all  the  circles  on  a 
sphere ;  of  all  the  circles  on  a  sphere  that  pass  through  a 
fixed  point ;  through  two  fixed  points ;  through  three  fixed 
points ;  of  all  the  circles  in  space ;  of  all  the  circles  that 
cut  across  a  given  line. 


ONE-TO-ONE  CORRESPONDENCE  13 

7.  Find,  the  order  of  infinity  of  all  lines  tangent  to  a 
sphere ;  of  all  planes  tangent  to  a  sphere ;  of  lines  and 
planes  tangent  to  a  sphere  and  passing  through  a  fixed  point. 

8.  Set  up  a  one-to-one  correspondence  between  the  series 
of  numbers  1,  2,  3,  4,  •  •  •  and  the  series  of  even  numbers 
2,  4,  6,  8  ■  •  •.  Are  we  justified  in  saying  that  there  are  just 
as  many  even  numbers  as  there  are  numbers  altogether  ? 

9.  Is  the  axiom  "  The  whole  is  greater  than  one  of  its 
parts  "  applicable  to  infinite  assemblages  ? 

10.  Make  out  a  classified  list  of  all  the  infinitudes  of  the 
first,  second,  third,  and  fourth  orders  mentioned  in  this 
chapter. 


CHAPTER  II 

RELATIONS   BETWEEN  FUNDAMENTAL  FORMS   IN  ONE- 
TO-ONE  CORRESPONDENCE  WITH  EACH  OTHER 

23.  Seven  fundamental  forms.  In  the  preceding  chap- 
ter we  have  called  attention  to  seven  fundamental  forms  : 
the  point-row,  the  pencil  of  rays,  the  axial  pencil,  the 
plane  system,  the  point  system,  the  space  system,  and 
the  system  of  lines  in  space.  These  fundamental  forms 
are  the  material  which  we  intend  to  use  in  building  up 
a  general  theory  which  will  be  found  to  include  ordinary 
geometry  as  a  special  case.  We  shall  be  concerned,  not 
with  measurement  of  angles  and  areas  or  line  seg- 
ments, as  in  the  study  of  Euclid,  but  in  combining  and 
comparing  these  fundamental  forms  and  in  "generating'' 
new  forms  by  means  of  them.  In  problems  of  con- 
struction we  shall  make  no  use  of  measurement,  either 
of  angles  or  of  segments,  and  except  in  certain  special 
applications  of  the  general  theory  we  shall  not  find  it 
necessary  to  require  more  of  ourselves  than  the  ability 
to  draw  the  line  joining  two  points,  or  to  find  the  point 
of  intersections  of  two  lines,  or  the  line  of  intersection 
of  two  planes,  or,  in  general,  the  common  elements  of 
two  fundamental  forms. 

24.  Projective  properties.  Our  chief  interest  in  this 
chapter  will  be  the  discovery  of  relations  between 
the   elements    of   one   form   which   hold    between   the 

14 


FUNDAMENTAL  FOEMS  15 

corresponding  elements  of  any  other  form  in  one-to-one 
correspondence  with  it.  We  have  already  called  atten- 
tion to  the  danger  of  assuming  that  whatever  relations 
hold  between  the  elements  of  one  assemblage  must  also 
hold  between  the  corresponding  elements  of  any  assem- 
blage in  one-to-one  correspondence  with  it.  This  false 
assumption  is  the  basis  of  the  so-called  "proof  by 
analogy"  so  much  in  vogue  among  speculative  theorists. 
When  it  appears  that  certain  relations  existing  between 
the  points  of  a  given  point-row  do  not  necessitate  the 
same  relations  between  the  corresponding  elements  of 
another  in  one-to-one  correspondence  with  it,  we  should 
view  with  suspicion  any  application  of  the  "  proof  by 
analogy "  in  realms  of  thought  where  accurate  judg- 
ments are  not  so  easily  made.  For  example,  if  in  a 
given  point-row  u  three  points,  A,  B,  and  C,  are  taken 
such  that  B  is  the  middle  point  of  the  segment  AC, 
it  does  not  follow  that  the  three  points  A\  B',  C 
in  a  point-row  perspective  to  u  will  be  so  related. 
Relations  between  the  elements  of  any  form  which  do 
go  over  unaltered  to  the  corresponding  elements  of 
a  form  projectively  related  to  it  are  called  projective 
relations.  Relations  involving  measurement  of  lines  or 
of  angles  are  not  projective. 

25.  Desargues's  theorem.  We  consider  first  the  fol- 
lowing beautiful  theorem,  due  to  Desargues  and  called 
by  his  name. 

If  two  triangles,  A,  B,  C  and  A',  B',  C,  are  so  situated 
that  the  lines  AA',  BB',  and  C'C  all  meet  in  a  point,  then 
the  pairs  of  sides  AB  and  A'B',  BC  and  B'C,  CA  and 
C'A'  all  meet  on  a  straight  lute,  and  conversely. 


1G 


PKOJECTIVE  GEOMETRY 


Fig.  :5 


Let  the  lines  AA',  BH',  and  C<"  meet  in  the  point  .1/ 
(Fig.  8).  Conceive  of  the  figure  as  in  space,  so  that 
M  is  the  vertex  of  a  trihedral  angle  of  which  the  given 
triangles  are  plane  sections.  The  lines  AB  and  A'B'  are 
in  the  same  plane  and  must  meet  when  produced,  their 
point  of  intersection 
being  clearly  a  point 
ill  the  plane  of  each 
triangle  and  there- 
fore in  the  line  of 
intersection  of  these 
two  planes.  Call  this 
point  P.  By  similar 
reasoning  the  point 
Q  of  intersection  of 
the  lines  BC  and 
B'C'  must  lie  on  this  same  line  as  well  as  the  point  22 
of  intersection  of  CA  and  C'A'.  Therefore  the  points 
P,  Q,  and  R  all  lie  on  the  same  line  m.  If  now  we  con- 
sider the  figure  a  plane  figure,  the  points  P,  Q,  and  R 
still  all  lie  on  a  straight  line,  which  proves  the  theorem. 
The  converse  is  established  in  the  same  manner. 

26.  Fundamental  theorem  concerning  two  complete 
quadrangles.  This  theorem  throws  into  our  hands  the 
following  fundamental  theorem  concerning  two  com- 
plete quadrangles,  a  complete  quadrangle  being  defined 
as  the  figure  obtained  by  joining  any  four  given  points 
by  straight  lines  in  the  six  possible  ways. 

Given  two  complete  quadrangles,  K,  L,  M,  X  and 
K',  L\  M\  X',  so  related  that  KL,  K'L',  MX,  M'X'  all 
meet  in  a  point  A  ;    LM,  L'M',  XK,  X' K'  all  meet  in  a 


FUNDAMENTAL   FORMS  17 

point  C;  and  LX,  L'N'  inert  in  a  point  B  on  ih<>  line 
A  C ;  then  the  lines  KM  and  K'M'  also  meet  in  a  point  I) 
on  the  line  AC. 

For,  by  the  converse  of  the  last  theorem,  KK',  LL', 
and  NN'  all  meet  in  a  point  S  (Fig.  4).  Also  LL',  MM, 
and  iVTV'  meet  in  a  point,  and  therefore  in  the  same 


point  &  Thus  A'A'',  LL',  and  MM'  meet  in  a  point, 
and  so,  by  Desargues's  theorem  itself,  A,  B,  and  D  are 
on  a  straight  line. 

27.  Importance  of  the  theorem.  The  importance  of 
this  theorem  lies  in  the  fact  that,  A,  B,  and  C  being 
given,  an  indefinite  number  of  quadrangles  A'',  L',  M',N' 
may  be  found  such  that  K'L'  and  M'N'  meet  in  A,  K'N' 
and  L'M'  in  C,  with  L'N'  passing  through  B.  Indeed, 
the  lines  A K'  and  AM'  may  be  drawn  arbitrarily 
through  A,  and  any  line  through  B  may  be  used  to 
determine  L'  and  N'.  By  joining  these  two  points  to 
C  the  points  K'  and  M'  are  determined.    Then  the  line 


18  PROJECTIVE  GEOMETRY 

joining  K'  and  M',  found  in  this  way,  must  pass 
through  the  point  D  already  determined  by  the  quad- 
rangle K,  L,  M,  iV.  The  three  points  A,  B,  <\  given  M 
order,  serve  thus  to  determine  a  fourth  point  D. 

28.  In  a  complete  quadrangle  the  line  joining  any 
two  points  is  called  the  opposite  side  to  the  line  joining 
the  other  two  points.  The  result  of  the  preceding 
paragraph  may  then  be  stated  as  follows : 

Given  three  points,  A,  B,  C,  in  a  straight  line,  if  a 
pair  of  opposite  sides  of  a  complete  quadrangle  pass 
through  Ay  and  another  pair  through  C,  and  one  of  the 
remaining  two  sides  goes  through  B,  then  the  other  of 
the  remaining  two  sides  will  go  through  a  fixed  point 
which  does  not  depend  on  the  quadrangle  employed. 

29.  Four  harmonic  points.  Four  points,  A,  B,  C,  D, 
related  as  in  the  preceding  theorem  are  called  four 
harmonic  points.  The  point  D  is  called  the  fourth  har- 
monic of  B  with  respect  to  A  and  C.  Since  B  and  I)  play 
exactly  the  same  role  in  the  above  construction,  B  is 
also  the  fourth  harmonic  of  D  with  respect  to  A  and  C. 
B  and  D  are  called  harmonic  conjugates  with  respect  to 
A  and  C.  We  proceed  to  show  that  A  and  C  are  also 
harmonic  conjugates  with  respect  to  B  and  D  —  that  is, 
that  it  is  possible  to  find  a  quadrangle  of  which  two 
opposite  sides  shall  pass  through  B,  two  through  D, 
and  of  the  remaining  pair,  one  through  A  and  the  other 
through  C. 

Let  O  be  the  intersection  of  KM  and  LN  (Fig.  5). 
Join  0  to  A  and  C.  The  joining  lines  cut  out  on  the 
sides  of  the  quadrangle  four  points,  F,  Q,  i**,  S.  ( lonsidex 
the  quadrangle  P,  A',  Q,  0.    One  pair  of  opposite  sides 


FUNDAMENTAL  FORMS 


19 


Fig 


passes  through  A,  one  through  C,  and  one  remaining  side 
through  D;  therefore  the  other  remaining  side  must 
pass  through  B.  Similarly,  RS  passes  through  B  and 
PS  and  QB  pass 
through  I).  The 
quadrangle  P,  Q, 
B,  S  therefore 
lias  two  opposite 
sides  through  B, 
two  through  D, 
and  the  remain- 
ing pair  through 
A  and  C.   A  and 

C  are  thus  harmonic  conjugates  with  respect  to  B  and  D. 
We  may  sum  up  the  discussion,  therefore,  as  follows : 

30.  If  A  and  C  are  harmonic  conjugates  with  respect 
to  B  and  D,  then  B  and  D  are  harmonic  conjugates  with 
respect  to  A  and  C. 

31.  Importance  of  the  notion.  The  importance  of  the 
notion  of  four  harmonic  points  lies  in  the  fact  that  it 
is  a  relation  which  is  carried  over  from  four  points  in 
a  point-row  u  to  the  four  points  that  correspond  to 
them  in  any  point-row  u!  perspective  to  u. 

To  prove  this  statement  we  construct  a  quadrangle 
K,  L,  M,  AT  such  that  KL  and  MN  pass  through  A,  KN 
and  LM  through  C,  LN  through  B,  and  KM  through  D. 
Take  now  any  point  S  not  in  the  plane  of  the  quad- 
rangle and  construct  the  planes  determined  by  S  and 
all  the  seven  lines  of  the  figure.  Cut  across  this  set  of 
planes  by  another  plane  not  passing  through  &  This 
plane   cuts    out    on    the    set    of   seven   planes    another 


20  PROJECTIVE  GEOMETRY 

quadrangle  which  determines  four  new  harmonic  points, 
A',  B',  C,  D',  on  the  lines  joining  S  to  A,  B,  C,  D.  But 
S  may  be  taken  as  any  point,  since  the  original  quad- 
rangle may  be  taken  in  any  plane  through  A,  B,  C,  D; 
and,  further,  the  points  A',  B',  C',  D'  are  the  intersection 
of  SA,  SB,  SC,  SD  by  any  line.  We  have,  then,  the 
remarkable  theorem: 

32.  If  any  point  is  joined  to  four  harmonic  points,  and 
the  four  lines  thus  obtained  are  cut  by  any  fifth,  the  four 
points  of  intersection  are  again  harmonic. 

33.  Four  harmonic  lines.  We  are  now  able  to  extend 
the  notion  of  harmonic  elements  to  pencils  of  rays,  and 
indeed  to  axial  pencils.  For  if  we  define  four  harmonic 
rays  as  four  rays  which  pass  through  a  point  and  which 
pass  one  through  each  of  four  harmonic  points,  we  have 
the  theorem 

Four  harmonic  lines  are  cut  by  any  transversal  in  four 
harmonic  points. 

34.  Four  harmonic  planes.  We  also  define  four  har- 
monic planes  as  four  planes  through  a  line  which  pass 
one  through  each  of  four  harmonic  points,  and  we  may 
show  that 

Four  harmonic  planes  are  cut  by  any  plane  not  passing 
through  their  common  line  in  four  harmonic  lines,  and  also 
by  any  line  in  four  harmonic  points. 

For  let  the  planes  a,  /3,  7,  8,  which  all  pass  through 
the  line  g,  pass  also  through  the  four  harmonic  points 
A,  B,  C,  D,  so  that  a  passes  through  A,  etc.  Then  it  is 
clear  that  any  plane  ir  through  A,  B,  C,  D  will  cut  out 
four  harmonic  lines  from  the  four  planes,  for  they  are 


FUNDAMENTAL  FOEMS  21 

lines  through  the  intersection  P  of  g  with  the  plane 
7r,  and  they  pass  through  the  given  harmonic  points 
A,  B,  C,  D.  Any  other  plane  a  cuts  g  in  a  point  S  and 
cuts  a,  /3,  7,  8  in  four  lines  that  meet  ir  in  four  points 
A',  B',  C",  D'  lying  on  PA,  PB,  PC,  and  PD  respec- 
tively, and  are  thus  four  harmonic  lines.  Further,  any 
ray  cuts  a,  yS,  7,  8  in  four  harmonic  points,  since  any 
plane  through  the  ray  gives  four  harmonic  lines  of 
intersection. 

35.  These  results  may  be  put  together  as  follows : 

Given  any  two  assemblages  of  points,  rags,  or  planes, 
perspectively  related  to  each  other,  four  harmonic  elements 
of  one  must  correspond  to  four  elements  of  the  other  which 
are  likewise  harmonic. 

If,  now,  two  forms  are  perspectively  related  to  a  thircl, 
any  four  harmonic  elements  of  one  must  correspond  to 
four  harmonic  elements  in  the  other.  We  take  this  as 
our  definition  of  projective  correspondence,  and  say: 

36.  Definition  of  projectivity.  Two  fundamental  forms 
are  protectively  related  to  each  other  when  a  one-to-one  cor- 
respondence exists  between  the  elements  of  the  two  and  when 
four  harmonic  elements  of  one  correspond  to  four  harmonic 
elements  of  the  other. 

37.  Correspondence  between  harmonic  conjugates.  Given 
four  harmonic  points,  A,  B,  C,  D;  if  we  fix  A  and  C, 
then  B  and  D  vary  together  in  a  way  that  should  be 
thoroughly  understood.  To  get  a  clear  conception  of 
their  relative  motion  we  may  fix  the  points  L  and  M  of 
the  quadrangle  K,  L,  M,  A"  (Fig.  6).  Then,  as  B  describes 
the  point-row  AG,  the  point  N  describes  the  point-row 


22 


PROJECTIVE  GEOMETRY 


Fit;.  6 


AM  perspective  to  it.  Projecting  N  again  from  C,  we 
get  a  point-row  K  on  AL  perspective  to  the  point-row 
N  and  thus  projective  to  the  point-row  B.  Project  the 
point-row  K  from  M  and  we  get  a  point-row  D  on 
AC  again,  which  is  projective  to  the  point-row  B.  For 
every  point  B  we  have  thus  one  and  only  one  point 
D,  and  conversely. 
In  other  words,  we 
have  set  up  a  one- 
to-one  correspond- 
ence between  the 
points  of  a  single 
point-row,  which  is 
also  a  projective 
correspondence  be- 
cause four  har- 
monic points  B  correspond  to  four  harmonic  points  D. 
We  may  note  also  that  the  correspondence  is  here  char- 
acterized by  a  feature  which  does  not  always  appear  in 
projective  correspondences :  namely,  the  same  process 
that  carries  one  from  B  to  D  will  carry  one  back  from 
rD  to  B  again.  This  special  property  will  receive  further 
study  in  the  chapter  on  Involution. 

38.  It  is  seen  that  as  B  approaches  A,  D  also  ap- 
proaches A.  As  B  moves  from  A  toward  C,  D  moves 
from  A  in  the  opposite  direction,  passing  through  the 
point  at  infinity  on  the  line  AC,  and  returns  on  the 
other  side  to  meet  B  at  C  again.  In  other  words,  as  B 
traverses  AC,  D  traverses  the  rest  of  the  line  from  A  to 
C  through  infinity.  In  all  positions  of  B,  except  at  A  or 
C,  B  and  D  are  separated  from  each  other  by  A  and  C. 


FUNDAMENTAL  FORMS 


23 


39.  Harmonic  conjugate  of  the  point  at  infinity.  It  is 
natural  to  inquire  what  position  of  B  corresponds  to  the 
infinitely  distant  position  of  D.  We  have  proved  (§  27) 
that  the  particular  quadrangle  K,  L,  M,  N  employed  is 
of  no  consequence.  We  shall  therefore  avail  ourselves  of 
one  that  lends  itself  most  readily  to 
the  solution  of  the  problem.  We 
choose  the  point  L  so  that  the  trian- 
gle ALC  is  isosceles  (Fig.  7).  Since 
D  is  supposed  to  be  at  infinity,  the 
line  KM  is  parallel  to  AC.  There- 
fore the  triangles  KAC  and  MAC 
are  equal,  and  the  triangle  ANC  is  also  isosceles.  The 
triangles  CNL  and  ANL  are  therefore  equal,  and  the  line 
LB  bisects  the  angle  ALC.  B  is  therefore  the  middle 
point  of  AC,  and  Ave  have  the  theorem 

The  harmonic  conjugate  of  the  middle  point  of  AC  is  at 
infinity. 

40.  Projective  theorems  and  metrical  theorems.  Linear 
construction.  This  theorem  is  the  connecting  link  be- 
tween the  general  projective  theorems  which  we  have 
been  considering  so  far  and  the  metrical  theorems  of 
ordinary  geometry.  Up  to  this  point  we  have  said  noth- 
ing about  measurements,  either  of  line  segments  or  of 
angles.  Desargues's  theorem  and  the  theory  of  harmonic 
elements  which  depends  on  it  have  nothing  to  do  with 
magnitudes  at  all.  Not  until  the  notion  of  an  infinitely 
distant  point  is  brought  in  is  any  mention  made  of 
distances  or  directions.  We  have  been  able  to  make 
all  of  our  constructions  up  to  this  point  by  means  of 
the  straightedge,  or  ungraduated  ruler.    A  construction 


24  PROJECTIVE  GEOMETRY 

made  with  such  an  instrument  we  shall  call  a  linear 
construction.  It  requires  merely  that  we  be  able  to 
draw  the  line  joining  two  points  or  find  the  point  of 
intersection  of  two  lines. 

41.  Parallels  and  mid-points.  It  might  be  thought 
that  drawing  a  line  through  a  given  point  parallel  to 
a  given  line  was  only  a  special  case  of  drawing  a  line 
joining  two  points.  Indeed,  it  consists  only  in  draw- 
ing a  line  through  the  given  point  and  through  the 
"  infinitely  distant  point "  on  the  given  line.  It  must 
be  remembered,  however,  that  the  expression  "infinitely 
distant  point"  must  not  be  taken  literally.  When  we 
say  that  two  parallel  lines  meet  "  at  infinity,"  we  really 
mean  that  they  do  not  meet  at  all,  and  the  only  reason 
for  using  the  expression  is  to  avoid  tedious  statement 
of  exceptions  and  restrictions  to  our  theorems.  We 
ought  therefore  to  consider  the  drawing  of  a  line  par- 
allel to  a  given  line  as  a  different  accomplishment  from 
the  drawing  of  the  line  joining  two  given  points.  It  is 
a  remarkable  consequence  of  the  last  theorem  that  a 
parallel  to  a  given  line  and  the  mid-point  of  a  given 
segment  are  equivalent  data.  For  the  construction  is 
reversible,  and  if  we  are  given  the  middle  point  of  a 
given  segment,  we  can  construct  linearly  a  line  parallel  to 
that  segment.  Thus,  given  that  B  is  the  middle  point  of 
AC,  we  may  draw  any  two  lines  through  A,  and  any  line 
through  B  cutting  them  in  points  X  and  L.  Join  N  and 
L  to  C  and  get  the  points  K  and  M  on  the  two  lines 
through  A.  Then  KM  is  parallel  to  AC.  The  bisection  of 
a  given  segment  and  the  drawing  of  a  line  parallel  to  the 
segment  are  equivalent  data  when  linear  construction  is  used. 


FUNDAMENTAL  FORMS  25 

42.  It  is  not  difficult  to  give  a  linear  construction 
for  the  problem  to  divide  a  given  segment  into  n  equal 
parts,  given  only  a  parallel  to  the  segment.  This  is 
simple  enough  when  n  is  a  power  of  2.  For  any  other 
number,  such  as  29,  divide  any  segment  on  the  line 
parallel  to  AC  into  32  equal  parts,  by  a  repetition  of 
the  process  just  described.  Take  29  of  these,  and  join 
the  first  to  A  and  the  last  to  C.  Let  these  joining  lines 
meet  in  S.  Join  S  to  all  the  other  points.  Other 
problems,  of  a  similar  sort,  are  given  at  the  end  of 
the  chapter. 

43.  Numerical  relations.  Since  three  points,  given  in 
order,  are  sufficient  to  determine  a  fourth,  as  explained 
above,  it  ought  to  be  possible  to  reproduce  the  process 
numerically  in  view  of  the  one-to-one  correspondence 
which  exists  between  points  on  a  line  and  numbers;  a 
correspondence  which,  to  be  sure,  we  have  not  estab- 
lished here,  but  which  is  discussed  in  any  treatise 
on  the  theory  of  point  sets.  We  proceed  to  discover 
what  relation  between  four  numbers  corresponds  to  the 
harmonic  relation  between 
four   points. 

44.  Let  A,  B,  C,  D  be  four 
harmonic  points  (Fig.  8),  and 
let  SA,  SB,  SC,  SD  be  four 
harmonic  lines.  Assume  a 
line  drawn  through  B  parallel  Fig.  g 
to  SD,  meeting  SA  in  A'  and 

SC  in  C".  Then  A,'  B,  C,  and  the  infinitely  distant 
point  on  A'C  are  four  harmonic  points,  and  therefore 
B  is  the  middle  point  of  the  segment  A'C.    Then,  since 


26  PROJECTIVE  GEOMETRY 

the  triangle  DAS  is  similar  to  the  triangle  BAA',  we 
may  write  the  proportion 

AB:AD=BA':SD. 
Also,  from  the  similar  triangles  DSC  and  BCC,  we  have 

CD:CB  =  SD:BCf. 
From  these  two  proportions  we  have,  remembering  that 
BA'  =  BC,  AB-CD_     1 

AD-CB~       ' 
the  minus  sign  being  given  to  the  ratio  on  account  of  the 
fact  that  A  and  C  are  always  separated  from  B  and  D, 
so  that  one  or  three  of  the  segments  AB,  CD,  AD,  CB 
must  be  negative. 

45.  Writing  the  last  equation  in  the  form 
CB:AB  =  -CD:AD, 
and  using  the  fundamental  relation  connecting  three 
points  on  a  line,  pR  +  RQ  =  pQy 

which  holds  for  all  positions  of  the  three  points  if 
account  be  taken  of  the  sign  of  the  segments,  the  last 
proportion  may  be  written 

(CA  -  BA) :  AB  =  -  ( CA  -  DA) :  AD, 
or  (AB-AC):AB  =  (AC-AD):AD; 

so  that  AB,  AC,  and  AD  are  three  quantities  in  har- 
monic progression,  since  the  difference  between  the  first 
and  second  is  to  the  first  as  the  difference  between  the 
second  and  third  is  to  the  third.  Also,  from  this  last 
proportion  comes  the  familiar  relation 

2/AC  =  l/AB+l/AD, 
which  is  convenient  for  the  computation  of  the  distance 
AD  when  AB  and  AC  are  given  numerically. 


FUNDAMENTAL  FORMS  27 

46.  Anharmonic  ratio.  The  corresponding  relations 
between  the  trigonometric  functions  of  the  angles  deter- 
mined by  four  harmonic  lines  are  not  difficult  to  obtain, 
but  as  we  shall  not  need  them  in  building  up  the 
theory  of  projective  geometry,  we  will  not  discuss  them 
here.  Students  who  have  a  slight  acquaintance  with 
trigonometry  may  read  in  a  later  chapter  (§  161)  a 
development  of  the  theory  of  a  more  general  relation, 
called  the  anharmonic  ratio,  or  cross  ratio,  which  connects 
any  four  points  on  a  line. 

PROBLEMS 

1.  Draw  through  a  given  point  a  line  which  shall  pass 
through  the  inaccessible  point  of  intersection  of  two  given 
lines.  The  following  construction  may  be  made  to  depend 
upon  Desargues's  theorem  :  Through  the  given  point  P  draw 
any  two  rays  cutting  the  two  lines  in  the  points  AB'  and 
A'B,  A,  B,  lying  on  one  of  the  given  lines  and  A',  B',  on  the 
other.  Join  A  A'  and  BB',  and  find  their  point  of  intersec- 
tion S.  Through  S  draw  any  other  ray,  cutting  the  given 
lines  in  CC'.  Join  BC'  and  B'C,  and  obtain  their  point 
of  intersection  Q.  PQ  is  the  desired  line.  Justify  this 
construction. 

2.  To  draw  through  a  given  point  P  a  line  which  shall 
meet  two  given  lines  in  points  A  and  B,  equally  distant  from 
P.  Justify  the  following  construction  :  Join  P  to  the  point 
S  of  intersection  of  the  two  given  lines.  Construct  the 
fourth  harmonic  of  PS  with  respect  to  the  two  given  lines. 
Draw  through  P  a  line  parallel  to  this  line.  This  is  the 
required  line. 

3.  Given  a  parallelogram  in  the  same  plane  with  a  given 
segment  A  C,  to  construct  linearly  the  middle  point  of  A  C. 


28  PROJECTIVE  GEOMETRY 

4.  Given  four  harmonic  lines,  of  which  one  pair  are  at 
right  angles  to  each  other,  show  that  the  other  pair  make 
equal  angles  with  them.  This  is  a  theorem  of  which  frequent 
use  will  be  made. 

5.  Given  the  middle  point  of  a  line  segment,  to  draw  a 
line  parallel  to  the  segment  and  passing  through  a  given 
point. 

6.  A  line  is  drawn  cutting  the  sides  of  a  triangle  ABC  in 
the  points  A',  B',  C,  the  point  A'  lying  on  the  side  BC,  etc. 
The  harmonic  conjugate  of  .4'  with  respect  to  B  and  C  is 
then  constructed  and  called  A".  Similarly,  B"  and  C"  are 
constructed.  Show  that  A  "  B'  C"  lie  on  a  straight  line.  Find 
other  sets  of  three  points  on  a  line  in  the  figure.  Find  also 
sets  of  three  lines  through  a  point. 


CHAPTER  III 

COMBINATION   OF  TWO   PROJECTIVELY   RELATED 
FUNDAMENTAL  FORMS 

47.  Superposed  fundamental  forms.  Self-corresponding 
elements.  We  have  seen  (§37)  that  two  projective 
point-rows  may  be  superposed  upon  the  same  straight 
line.  This  happens,  for  example,  when  two  pencils 
which  are  projective  to  each  other  are  cut  across  by 
a  straight  line.  It  is  also  possible  for  two  projective 
pencils  to  have  the  same  center.  This  happens,  for 
example,  when  two  projective  point-rows  are  projected 
to  the  same  point.  Similarly,  two  projective  axial  pen- 
cils may  have  the  same  axis.  We  examine  now  the 
possibility  of  two  forms  related  in  this  way,  having 
an  element  or  elements  that  correspond  to  themselves. 
We  have  seen,  indeed,  that  if  B  and  D  are  harmonic 
conjugates  with  respect  to  A  and  C,  then  the  point- 
row  described  by  B  is  projective  to  the  point-row  de- 
scribed by  D,  and  that  A  and  C  are  self-corresponding 
points.  Consider  more  generally  the  case  of  two  pencils 
perspective  to  each  other  with  axis  of  perspectivity  u' 
(Fig.  9).  Cut  across  them  by  a  line  u.  We  get  thus 
two  projective  point-rows  superposed  on  the  same  line 
u,  and  a  moment's  reflection  serves  to  show  that  the 
point  N  of  intersection  u  and  u'  corresponds  to  itself 
in   the   two   point-rows.    Also,   the   point  Jf,  where   u 

29 


30 


PROJECTIVE  GEOMETRY 


Fig.  9 


intersects  the  line  joining  the  centers  of  the  two  pen- 
cils, is  seen  to  correspond  to  itself.  It  is  thus  possible 
for  two  projective  point- 
rows,  superposed  upon 
the  same  line,  to  have  two 
self-corresponding  points. 
Clearly  M  and  N  may 
fall  together  if  the  line 
joining  the  centers  of  the 
pencils  happens  to  pass 
through  the  point  of  in- 
tersection of  the  lines  u 
and  u'. 

48.  We  may  also  give  an  illustration  of  a  case 
where  two  superposed  projective  point-rows  have  no 
self-corresponding  points  at  all.  Thus  we  may  take 
two  lines  revolving  about  a  fixed 
point  S  and  always  making  the 
same  angle  a  with  each  other 
(Fig.  10).  They  will  cut  out  on 
any  line  u  in  the  plane  two  point- 
rows  which  are  easily  seen  to  be 
projective.  For,  given  any  four 
rays  SP  which  are  harmonic,  the 
four  corresponding  rays  SP'  must 
also  be  harmonic,  since  they  make 
the  same  angles  with  each  other. 
Four  harmonic  points  P  corre- 
spond, therefore,  to  four  harmonic  points  P'.  It  is  clear, 
however,  that  no  point  P  can  coincide  with  its  corre- 
sponding point  P',  for  in  that  case  the  lines  PS  and 


Fir..  10 


TWO  FUNDAMENTAL  FORMS  31 

P'S  would  coincide,  which  is  impossible  if  the  angle 
between  them  is  to  be  constant. 

49.  Fundamental  theorem.  Postulate  of  continuity. 
We  have  thus  shown  that  two  projective  point-rows, 
superposed  one  on  the  other,  may  have  two  points,  one 
point,  or  no  point  at  all  corresponding  to  themselves. 
We  proceed  to  show  that 

If  two  projective  point-rows,  superposed  upon  the  same 
straight  line,  have  more  than  two  self-corresponding  points, 
they  must  have  an  infinite  number,  and  every  point  corre- 
sponds to  itself ;  that  is,  the  two  point-rows  are  not 
essentially  distinct. 

If  three  points,  A,  B,  and  C,  are  self-corresponding, 
then  the  harmonic  conjugate  D  of  B  with  respect  to  A 
and  C  must  also  correspond  to  itself.  For  four  harmonic 
points  must  always  correspond  to  four  harmonic  points. 
In  the  same  way  the  harmonic  conjugate  of  D  with 
respect  to  B  and  C  must  correspond  to  itself.  Combining 
new  points  with  old  in  this  way,  we  may  obtain  as  many 
self-corresponding  points  as  we  wish.  We  show  further 
that  every  point  on  the  line  is  the  limiting  point  of  a 
finite  or  infinite  sequence  of  self-corresponding  points. 
Thus,  let  a  point  P  lie  between  A  and  B.  Construct 
now  D,  the  fourth  harmonic  of  C  with  respect  to  A  and 
B.  D  may  coincide  with  P,  in  which  case  the  sequence 
is  closed ;  otherwise  P  lies  in  the  stretch  AD  or  in  the 
stretch  DB.  If  it  lies  in  the  stretch  DB,  construct  the 
fourth  harmonic  of  C  with  respect  to  D  and  B.  This 
point  D'  may  coincide  with  P,  in  which  case,  as  before, 
the  sequence  is  closed.  If  P  lies  in  the  stretch  DD', 
we  construct  the   fourth   harmonic   of    C  with  respect 


32  PROJECTIVE  GEOMETRY 

to  DD',  etc.  In  each  step  the  region  in  which  P  lies  is 
diminished,  and  the  process  may  be  continued  until  two 
self-corresponding  points  are  obtained  on  either  side  of 
P,  and  at  distances  from  it  arbitrarily  small. 

We  now  assume,  explicitly,  the  fundamental  postulate 
that  the  correspondence  is  continuous,  that  is,  that  the 
distance  between  two  points  in  one  point-row  may  be  made 
arbitrarily  small  by  sufficiently  diminishing  the  distance 
between  the  corresponding  points  in  the  other.  Suppose 
now  that  P  is  not  a  self-corresponding  point,  but  cor- 
responds to  a  point  P'  at  a  fixed  distance  d  from  P. 
As  noted  above,  we  can  find  self-corresponding  points 
arbitrarily  close  to  P,  and  it  appears,  then,  that  we  can 
take  a  point  D  as  close  to  P  as  we  wish,  and  yet  the 
distance  between  the  corresponding  points  I)'  and  P' 
approaches  d  as  a  limit,  and  not  zero,  which  contradicts 
the  postulate  of  continuity. 

50.  It  follows  also  that  two  projective  pencils  which 
have  the  same  center  may  have  no  more  than  two  self- 
corresponding  rays,  unless  the  pencils  are  identical.  For 
if  we  cut  across  them  by  a  line,  we  obtain  two  projec- 
tive point-rows  superposed  on  the  same  straight  line, 
which  may  have  no  more  than  two  self-corresponding 
points.  The  same  considerations  apply  to  two  projective 
axial  pencils  which  have  the  same  axis. 

51.  Projective  point-rows  having  a  self -corresponding 
point  in  common.  Consider  now  two  projective  point- 
rows  lying  on  different  lines  in  the  same  plane.  Their 
common  point  may  or  may  not  be  a  self-corresponding 
point.  If  the  two  point-rows  are  perspectively  related, 
then  their  common  point  is  evidently  a  self -corresponding 


TWO  FUNDAMENTAL  FORMS  33 

point.  The  converse  is  also  true,  and  we  have  the  very 
important  theorem : 

52.  If  in  two  projective  point-rows  the  point  of  inter- 
section corresponds  to  itself  then  the  point-rotvs  are  in 
perspective  position. 

Let  the  two  point-rows  be  u  and  u'  (Fig.  11).  Let 
A  and  A',  B  and  B',  be  corresponding  points,  and  let 
also  the  point  M  of  intersection  of  u  and  u'  correspond 
to  itself.  Let  AA'  and  BB'  meet  in  the  point  S.  Take 
S  as  the  center  of  two  pencils, 
one  perspective  to  u  and  the  other 
perspective  to  u'.  In  these  two 
pencils  SA  coincides  with  its  cor- 
responding ray  SA',  SB  with  its 
corresponding  ray  S"B',  and  SM  /  p  ^ 
with  its  corresponding  ray  SM'. 

The  two  pencils  are  thus  identical,  by  the  preceding 
theorem,  and  any  ray  SD  must  coincide  with  its  cor- 
responding ray  SD'.  Corresponding  points  of  u  and  u', 
therefore,  all  lie  on  lines  through  the  point  S. 

53.  An  entirely  similar  discussion  shows  that 

If  in  two  projective  pencils  the  line  joining  their  cen- 
ters is  a  self-corresponding  ray,  then  the  two  pencils  are 
perspectively  related. 

54.  A  similar  theorem  may  be  stated  for  two  axial 
pencils  of  which  the  axes  intersect.  Very  frequent  use 
will  be  made  of  these  fundamental  theorems. 

55.  Point-row  of  the  second  order.  The  question  nat- 
urally arises,  What  is  the  locus  of  points  of  intersec- 
tion  of   corresponding   rays   of  two   projective   pencils 


34  PROJECTIVE  GEOMETRY 

which  are  not  in  perspective  position  ?  This  locus, 
which  will  be  discussed  in  detail  in  subsequent  chapters, 
is  easily  seen  to  have  at  most  two  points  in  common 
with  any  line  in  the  plane,  and  on  account  of  this 
fundamental  property  will  be  called  a  point-row  of  the 
second  order.  For  any  line  u  in  the  plane  of  the  two 
pencils  will  be  cut  by  them  in  two  projective  point- 
rows  which  have  at  most  two  self-corresponding  points. 
Such  a  self-corresponding  point  is  clearly  a  point  of 
intersection  of  corresponding  rays  of  the  two  pencils. 

56.  This  locus  degenerates  in  the  case  of  two  per- 
spective pencils  to  a  pair  of  straight  lines,  one  of  which 
is  the  axis  of  perspectivity  and  the  other  the  common 
ray,  any  point  of  which  may  be  considered  as  the  point 
of  intersection  of  corresponding  rays  of  the  two  pencils. 

57.  Pencils  of  rays  of  the  second  order.  Similar  inves- 
tigations may  be  made  concerning  the  system  of  lines 
joining  corresponding  points  of  two  projective  point- 
rows.  If  we  project  the  point-rows  to  any  point  in  the 
plane,  we  obtain  two  projective  pencils  having  the  same 
center.  At  most  two  pairs  of  self-corresponding  rays 
may  present  themselves.  Such  a  ray  is  clearly  a  line 
joining  two  corresponding  points  in  the  two  point-rows. 
The  result  may  be  stated  as  follows :  Tlie  system  of  rays 
joining  corresponding  points  in  two  projective  point-rows 
has  at  most  two  rays  in  common  with  any  pencil  in  the 
plane.  For  that  reason  the  system  of  rays  is  called  a 
pencil  of  rays  of  the  second  order. 

58.  In  the  case  of  two  perspective  point-rows  this 
system  of  rays  degenerates  into  two  pencils  of  rays  of 
the  first  order,  one  of  which  has  its  center  at  the  center 


TWO  FUNDAMENTAL  FORMS  35 

of  perspectivity  of  the  two  point-rows,  and  the  other  at 
the  intersection  of  the  two  point-rows,  any  ray  through 
which  may  be  considered  as  joining  two  corresponding 
points  of  the  two  point-rows. 

59.  Cone  of  the  second  order.  The  corresponding 
theorems  in  space  may  easily  be  obtained  by  joining 
the  points  and  lines  considered  in  the  plane  theorems 
to  a  point  S  in  space.  Two  projective  pencils  give  rise 
to  two  projective  axial  pencils  with  axes  intersecting. 
Corresponding  planes  meet  in  lines  which  all  pass 
through  S  and  through  the  points  on  a  point-row  of 
the  second  order  generated  by  the  two  pencils  of  rays. 
They  are  thus  generating  lines  of  a  cone  of  the  second 
order,  or  quadric  cone,  so  called  because  every  plane  in 
space  not  passing  through  S  cuts  it  in  a  point-row  of 
the  second  order,  and  every  line  also  cuts  it  in  at  most 
two  points.  If,  again,  we  project  two  point-rows  to  a 
point  S  in  space,  we  obtain  two  pencils  of  rays  with  a 
common  center  but  lying  in  different  planes.  Corre- 
sponding lines  of  these  pencils  determine  planes  which 
are  the  projections  to  S  of  the  lines  which  join  the  cor- 
responding points  of  the  two  point-rows.  At  most  two 
such  planes  may  pass  through  any  ray  through  S.  It 
is  called  a  pencil  of  planes  of  the  second  order. 

PROBLEMS 

1.  A  man  A  moves  along  a  straight  road  u,  and  another 
man  B  moves  along  the  same  road  and  walks  so  as  always 
to  keep  sight  of  A  in  a  small  mirror  M  at  the  side  of  the 
road.  How  many  times  will  they  come  together,  A  moving 
always  in  the  same  direction  along  the  road  ? 


36  PROJECTIVE  GEOMETRY 

2.  How  many  times  would  the  two  men  in  the  first  prob- 
lem see  each  other  in  two  mirrors  M  and  N  as  they  walk 
along  the  road  as  before  ?  (The  planes  of  the  two  mirrors 
are  not  necessarily  parallel  to  u.) 

3.  As  A  moves  along  u,  trace  the  path  of  B  so  that  the 
two  men  may  always  see  each  other  in  the  two  mirrors. 

4.  Two  boys  walk  along  two  paths  u  and  u\  each  holding 
a  string  which  they  keep  stretched  tightly  between  them. 
They  both  move  at  constant  but  different  rates  of  speed, 
letting  out  the  string  or  drawing  it  in  as  they  walk.  How 
many  times  will  the  line  of  the  string  pass  over  any  given 
point  in  the  plane  of  the  paths  ? 

5.  Trace  the  lines  of  the  string  when  the  two  boys  move 
at  the  same  rate  of  speed  in  the  two  paths  but  do  not  start 
at  the  same  time  from  the  point  where  the  two  paths 
intersect. 

6.  A  ship  is  sailing  on  a  straight  course  and  keeps  a  gun 
trained  on  a  point  on  the  shore.  Show  that  a  line  at  right 
angles  to  the  direction  of  the  gun  at  its  muzzle  will  pass 
through  any  point  in  the  plane  twice  or  not  at  all.  (Con- 
sider the  point-row  at  infinity  cut  out  by  a  line  through  the 
point  on  the  shore  at  right  angles  to  the  direction  of 
the  gun.) 

7.  Two  lines  u  and  u'  revolve  about  two  points  U  and  U' 
respectively  in  the  same  plane.  They  go  in  the  same  direc- 
tion and  at  the  same  rate  of  speed,  but  one  has  an  angle  a 
the  start  of  the  other.  Show  that  they  generate  a  point-row 
of  the  second  order. 

8.  Discuss  the  question  given  in  the  last  problem  when 
the  two  lines  revolve  in  opposite  directions.  Can  you 
recognize  the  locus  ? 


CHAPTER  IV 

POINT-ROWS  OF  THE  SECOND  ORDER 

60.  Point-row  of  the  second  order  defined.  We  have 
seen  that  two  fundamental  forms  in  one-to-one  corre- 
spondence may  sometimes  generate  a  form  of  higher 
order.  Thus,  two  point-rows  (§  55)  generate  a  system  of 
rays  of  the  second  order,  and  two  pencils  of  rays  (§  57), 
a  system  of  points  of  the  second  order.  As  a  system  of 
points  is  more  familiar  to  most  students  of  geometry 
than  a  system  of  lines,  we  study  first  the  point-row  of 
the  second  order. 

61.  Tangent  line.  We  have  shown  in  the  last  chapter 
(§  55)  that  the  locus  of  intersection  of  corresponding 
rays  of  two  projective  pencils  is  a  point-row  of  the 
second  order ;  that  is,  it  has  at  most  two  points  in  com- 
mon with  any  line  in  the  plane.  It  is  clear,  first  of  all, 
that  the  centers  of  the  pencils  are  points  of  the  locus ; 
for  to  the  line  SS',  considered  as  a  ray  of  S,  must 
correspond  some  ray  of  S'  which  meets  it  in  S'.  S', 
and  by  the  same  argument  S,  is  then  a  point  where 
corresponding  rays  meet.  Any  ray  through  S  will  meet 
it  in  one  point  besides  S,  namely,  the  point  P  where 
it  meets  its  corresponding  ray.  Now,  by  choosing  the 
ray  through  S  sufficiently  close  to  the  ray  SS',  the  point 
P  may  be  made  to  approach  arbitrarily  close  to  S',  and 
the  ray  S'P  may  be  made  to  differ  in  position  from  the 

37 


,69303 


38  PROJECTIVE  GEOMETRY 

tangent  line  at  S'  by  as  little  as  we  please.  We  have, 
then,  the  important  theorem 

The  ray  at  S'  which  corresponds  to  the  common  ray  SS' 
is  tangent  to  the  locus  at  S'. 

In  the  same  manner  the  tangent  at  S  may  be 
constructed. 

62.  Determination  of  the  locus.  We  now  show  that 
it  is  possible  to  assign  arbitrarily  the  position  of  three 
points,  A,  B,  and  C,  on  the  locus  (besides  the  points  S 
and  $')  ;  but,  these  three  points  being  chosen,  the  locus  is 
completely  determined. 

63.  This  statement  is  equivalent  to  the  following: 
Given  three  pairs  of  corresponding  rays  in  two  projective 

pencils,  it  is  possible  to  find  a  ray  of  one  which  corre- 
sponds to  any  ray  of  the  other. 

64.  We  proceed,  then,  to  the  solution  of  the  funda- 
mental 

Problem  :  Given  three  pairs  of  rays,  aa',  bb',  and  cc', 
of  two  projective  pencils,  S  and  S',  to  find  the  ray  d'  of  S' 
which  corresponds  to  any  ray  d  of  S. 

Call  A  the  intersection  of  aa',  B  the  intersection  of  bb', 
and  C  the  intersection  of  cc'  (Fig.  12).  Join  AB  by  the 
line  u,  and  AC  by  the  line  u'.  Consider  u  as  a  point- 
row  perspective  to  S,  and  u'  as  a  point-row  perspective 
to  S'.  u  and  u'  are  projectively  related  to  each  other, 
since  S  and  S'  are,  by  hypothesis,  so  related.  But  their 
point  of  intersection  A  is  a  self -corresponding  point,  since 
a  and  a'  were  supposed  to  be  corresponding  rays.  It  fol- 
lows (§52)  that  u  and  u'  are  in  perspective  position, 
and  that   lines  through  corresponding  points    all  pass 


POINT-ROWS  OF  THE  SECOND  ORDER       39 

through  a  point  M,  the  center  of  perspectivity,  the 
position  of  which  will  be  determined  by  any  two  such 
lines.  But  the  intersection  of  e  with  u  and  the  intersec- 
tion of  c'  with  u'  are  corresponding  points  on  u  and  u', 
and  the  line  joining  them  is  clearly  c  itself.  Similarly, 
b'  joins  two  corresponding  points  on  u  and  u',  and  so  the 
center  M  of  perspectivity  of  u  and  u'  is  the  intersection 


Fig.  12 

of  c  and  b'.  To  find  d'  in  S'  corresponding  to  a  given 
line  d  of  S,  we  note  the  point  L  where  d  meets  u.  Join 
L  to  M  and  get  the  point  N  where  this  line  meets  u'. 
L  and  N  are  corresponding  points  on  u  and  u',  and  d' 
must  therefore  pass  through  N.  The  intersection  D  of 
d  and  d'  is  thus  another  point  on  the  locus.  In  the  same 
manner  any  number  of  other  points  may  be  obtained. 

65.  The  lines  u  and  u'  might  have  been  drawn  in 
any  direction  through  A  (avoiding,  of  course,  the  line 
a  for  u  and  the  line  a'  for  w'),  and  the  center  of  per- 
spectivity M  would  be  easily  obtainable ;  but  the  above 
construction  furnishes  a  simple  and  instructive  figure. 
An  equally  simple  one  is  obtained  by  taking  a'  for  u 
and  a  for  u'. 


40  PROJECTIVE  GEOMETRY 

66.  Lines  joining  four  points  of  the  locus  to  a  fifth. 
Suppose  that  the  points  S,  S\  B,  C,  and  D  are  fixed, 
and  that  four  points,  A,  A^  A2,  and  Az,  are  taken  on  the 
locus  at  the  intersection  with  it  of  any  four  harmonic 
rays  through  B.  These  four  harmonic  rays  give  four 
harmonic  points,  L,  L^  etc.,  on  the  fixed  ray  SD.  These, 
in  turn,  project  through  the  fixed  point  M  into  four 
harmonic  points,  iV,  iV,  etc.,  on  the  fixed  line  DS'. 
These  last  four  harmonic  points  give  four  harmonic 
rays  CA,  CA^  CA2,  CAS.  Therefore  the  four  points  A 
which  project  to  B  in  four  harmonic  rays  also  pro- 
ject to  C  m  four  harmonic  rays.  But  C  may  be  any 
point  on  the  locus,  and  so  we  have  the  very  important 
theorem, 

Four  points  which  are  on  the  locus,  and  which  project 
to  a  fifth  point  of  the  locus  in  four  harmonic  rays,  project 
to  any  point  of  the  locus  in  four  harmonic  rays. 

67.  The  theorem  may  also  be  stated  thus: 

The  locus  of  points  from  which  four  given  points  are 
seen  along  four  harmonic  rays  is  a  point-row  of  the  second 
order  through  them. 

68.  A  further  theorem  of  prime  importance  also 
follows : 

Any  tivo  points  on  the  locus  may  be  taken  as  the  centers 
of  two  projective  pencils  which  will  generate  the  locus. 

69.  Pascal's  theorem.  The  points  A,  B,  C,  D,  S,  and 
S'  may  thus  be  considered  as  chosen  arbitrarily  on  the 
locus,  and  the  following  remarkable  theorem  follows 
at  once. 


POINT-ROWS  OF  THE  SECOND  ORDER      41 

Given  six  points,  1,  2,  3,  4,  5,  6,  on  the  point-row  of 
the  second  order,  if  we  call 

L   the  intersection  of  12  with  45, 
M  the  intersection  of  23  with  56, 
N  the  intersection  of  34  with  61, 
then  L,  M,  and  N  are  on  a  straight  line. 

70.  To  get  the  notation  to  correspond  to  the  figure,  we 
may  take  (Fig.  13)  A  =  1,  B  =  2,  S'  =  3,  D  =  4,  S  =  5,  and 
C=  6.  If  we  make  A  =  1,  C=  2,  S=  3, 1)  =  4,  S'  =  5,  and 
B  =  6,  the  points  L  and  iV  are  interchanged,  but  the  line 
is  left  unchanged. 
It  is  clear  that  one 
point  may  be  named 
arbitrarily  and  the 
other  five  named  in 
5  !  =  120  different 
ways,  but  since,  as 
we  have  seen,  two 
different  assignments 
of  names  give  the 
same  line,  it  follows 
that  there  cannot  be 
more  than  60  differ- 
ent lines  LMN  obtained  in  this  way  from  a  given  set  of 
six  points.  As  a  matter  of  fact,  the  number  obtained  in 
this  way  is  in  general  60.  The  above  theorem,  which 
is  of  cardinal  importance  in  the  theory  of  the  point-row 
of  the  second  order,  is  due  to  Pascal  and  was  discovered 
by  him  at  the  age  of  sixteen.  It  is,  no  doubt,  the  most 
important  contribution  to  the  theory  of  these  loci  since 


Fig.  13 


42  PROJECTIVE  GEOMETRY 

the  days  of  Apollonius.  If  the  six  points  be  called  the 
vertices  of  a  hexagon  inscribed  in  the  curve,  then  the 
sides  12  and  45  may  be  appropriately  called  a  pair  of 
opposite  sides.  Pascal's  theorem,  then,  may  be  stated 
as  follows: 

TJie  three  pairs  of  opposite  sides  of  a  hexagon  inscribed  in 
a  point-row  of  the  second  order  meet  in  three  points  on  a  line. 

71.  Harmonic  points  on  a  point-row  of  the  second  order. 
Before  proceeding  to  develop  the  consequences  of  this 
theorem,  we  note  another  result  of  the  utmost  impor- 
tance for  the  higher  developments  of  pure  geometry, 
which  follows  from  the  fact  that  if  four  points  on  the 
locus  project  to  a  fifth  in  four  harmonic  rays,  they  will 
project  to  any  point  of  the  locus  in  four  harmonic  rays. 
It  is  natural  to  speak  of  four  such  points  as  four  har- 
monic points  on  the  locus,  and  to  use  this  notion  to 
define  projective  correspondence  between  point-rows  of 
the  second  order,  or  between  a  point-row  of  the  second 
order  and  any  fundamental  form  of  the  first  order. 
Thus,  in  particular,  the  point-row  of  the  second  order, 
er,  is  said  to  be  perspectively  related  to  the  pencil  S  when 
every  ray  on  S  goes  through  the  point  on  a  which 
corresponds  to  it. 

72.  Determination  of  the  locus.  It  is  now  clear  that 
five  points,  arbitrarily  chosen  in  the  plane,  are  sufficient 
to  determine  a  point-row  of  the  second  order  through 
them.  Two  of  the  points  may  be  taken  as  centers  of 
two  projective  pencils,  and  the  three  others  will  deter- 
mine three  pairs  of  corresponding  rays  of  the  pencils, 
and  therefore  all  pairs.    If  four  points  of  the  locus  are 


POINT-ROWS  OF  THE  SECOND  ORDER      43 

given,  together  with  the  tangent  at  one  of  them,  the 
locus  is  likewise  completely  determined.  For  if  the  point 
at  which  the  tangent  is  given  be  taken  as  the  center  S 
of  one  pencil,  and  any  other  of  the  points  for  $',  then, 
besides  the  two  pairs  of  corresponding  rays  determined 
by  the  remaining  two  points,  we  have  one  more  pair, 
consisting  of  the  tangent  at  S  and  the  ray  SS'.  Simi- 
larly, the  curve  is  determined  by  three  points  and  the 
tangents  at  two  of  them. 

73.  Circles  and  conies  as  point-rows  of  the  second  order. 
It  is  not  difficult  to  see  that  a  circle  is  a  point-row  of 
the  second  order.  Indeed,  take  any  point  S  on  the  circle 
and  draw  four  harmonic  rays  through  it.  They  will  cut 
the  circle  in  four  points,  which  will  project  to  any  other 
point  of  the  curve  in  four  harmonic  rays;  for,  by  the 
theorem  concerning  the  angles  inscribed  in  a  circle,  the 
angles  involved  in  the  second  set  of  four  lines  are 
the  same  as  those  in  the  first  set.  If,  moreover,  we  pro- 
ject the  figure  to  any  point  in  space,  we  shall  get  a  cone, 
standing  on  a  circular  base,  generated  by  two  projective 
axial  pencils  which  are  the  projections  of  the  pencils 
at  S  and  S'.  Cut  across,  now,  by  any  plane,  and  we  get 
a  conic  section  which  is  thus  exhibited  as  the  locus  of 
intersection  of  two  projective  pencils.  It  thus  appears 
that  a  conic  section  is  a  point-row  of  the  second  order. 
It  will  later  appear  that  a  point-row  of  the  second  order 
is  a  conic  section.  In  the  future,  therefore,  we  shall 
refer  to  a  point-row  of  the  second  order  as  a  conic. 

74.  Conic  through  five  points.  Pascal's  theorem  fur- 
nishes an  elegant  solution  of  the  problem  of  drawing  a 
conic  through  five  given  points.    To  construct  a  sixth 


44 


PROJECTIVE  GEOMETRY 


Fn;.  14 


point  on  the  conic,  draw  through  the  point  numbered  1 
an  arbitrary  line  (Fig.  14),  and  let  the  desired  point 
6  be  the  second  point  of  intersection 
of  this  line  with  the  conic.  The  point 
L  =  12  —  45  is  obtainable  at  once ;  also 
the  point  N=  34  -  61.  But  L  and  N 
determine  Pascal's  line,  and  the  in- 
tersection of  23  with  56  must  be  on 
this  line.  Intersect,  then,  the  line  LN 
with  23  and  obtain  the  point  M.  Join 
M  to  5  and  intersect  with  61  for  the  desired  point  6. 

75.  Tangent  to  a  conic.  If  two  points  of  Pascal's  hex- 
agon approach  coincidence,  then  the  line  joining  them 
approaches  as  a  limiting  position  the  tangent  line  at  that 
point.  Pascal's  theorem  thus  affords  a  ready  method  of 
drawing  the  tangent  line  to  a  conic 
at  a  given  point.  If  the  conic  is  de- 
termined by  the  points  1,  2,  3,  4,  5 
(Fig.  15),  and  it  is  desired  to  draw 
the  tangent  at  the  point  1,  we  may 
call  that  point  1,  6.  The  points 
L  and  M  are  obtained  as  usual, 
and  the  intersection  of  34  with  LJf 
gives  JV.  Join  N  to  the  point  1  for 
the  desired  tangent  at  that  point. 

76.  Inscribed  quadrangle.  Two  pairs  of  vertices  may 
coalesce,  giving  an  inscribed  quadrangle.  Pascal's  theo- 
rem gives  for  this  case  the  very  important  theorem 

Two  pairs  of  opposite  sides  of  any  quadrangle  inscribed 
in  a  conic  meet  on  a  straight  line,  upon  which  line  also 
intersect  the  two  pairs  of  tangents  at  the  opposite  vertices. 


Fig.  15 


POINT-ROWS  OF  THE  SECOND  ORDER      45 


Fig.  16 


For  let  the  vertices  be  A,  B,  C,  and  D,  and  call  the 
vertex  A  the  point  1,  6 ;  B,  the  point  2 ;  C,  the  point 
3, 4 ;  and  D,  the  point  5  (Fig.  16).  Pascal's  theorem  then 
indicates  that 
L  =  AB-CD, 
M=AD-BC, 
and  N,  which 
is  the  inter- 
section of  the 
tangents  at  A 
and  C,  are  all 
on  a  straight 
line  u.  But 
if  we  were  to 

call  A  the  point  2,  B  the  point  6,1,  C  the  point  5,  and 
D  the  point  4,  3,  then  the  intersection  P  of  the  tangents 
at  B  and  D  are  also  on  this  same 
line  u.  Thus  L,  M,  iV,  and  P  are 
four  points  on  a  straight  line. 
The  consequences  of  this  theorem 
are  so  numerous  and  important 
that  we  shall  devote  a  separate 
chapter  to  them. 

77.  Inscribed  triangle.  Finally, 
three  of  the  vertices  of  the  hex- 
agon may  coalesce,  giving  a  trian- 
gle inscribed  in  a  conic.  Pascal's 
theorem  then  reads  as  follows  (Fig.  17)  for  this  case: 

The  three  tangents  at  the  vertices  of  a  triangle  inscribed 
in  a  conic  meet  the  opposite  sides  in  three  points  on  a 
straight  line. 


Fig.  17 


46 


PROJECTIVE  GEOMETRY 


78.  Degenerate  conic.  If  we  apply  Pascal's  theorem 
to  a  degenerate  conic  made  up  of  a  pair  of  straight 
lines,  we  get  the 
following  theo- 
rem (Fig.  18) : 

If  three  points, 
A,  B,  C,  are 
chosen  on  one 
line,  and  three 
points,  A',  B', 
C,  are  chosen  on 
another,  then  the 
three  points  L=AB'-A'B,  N  =  BC-B'C,  M=CA'-C'A 
are  all  on  a  straight  line. 


Pig.  18 


PROBLEMS 

1.  In  Fig.  12,  select  different  lines  u  and  u'  and  find 
for  each  pair  the  center  of  perspectivity  M. 

2.  Given  four  points,  A,  B,  C,  D,  in  the  plane,  construct 
a  fifth  point  P  such  that  the  lines  PA,  PB,  PC,  PD  shall  be 
four  harmonic  lines. 

Suggestion.  Draw  a  line  a  through  the  point  A  such  that  the  four 
lines  a,  AB,  AC,  AD  are  harmonic.  Construct  now  a  conic  through 
A,  B,  C,  and  D  having  a  for  a  tangent  at  A. 

3.  Where  are  all  the  points  P,  as  determined  in  the 
preceding  question,  to  be  found  ? 

4.  Select  any  five  points  in  the  plane  and  draw  the  tan- 
gent to  the  conic  through  them  at  each  of  the  five  points. 

5.  Given  four  points  on  the  conic,  and  the  tangent  at  one  of 
them,  to  construct  the  conic.  ("  To  construct  the  conic  "  means 
here  to  construct  as  many  other  points  as  may  be  desired.) 


POINT-ROWS  OF  THE  SECOND  ORDER       47 

6.  Given  three  points  on  the  conic,  and  the  tangent  at 
two  of  them,  to  construct  the  conic. 

7.  Given  five  points,  two  of  which  are  at  infinity  in 
different  directions,  to  construct  the  conic.  (In  this,  and 
in  the  following  examples,  the  student  is  supposed  to  be 
able  to  draw  a  line  parallel  to  a  given  line.) 

8.  Given  four  points  on  a  conic  (two  of  which  are  at  in- 
finity and  two  in  the  finite  part  of  the  plane),  together  with 
the  tangent  at  one  of  the  finite  points,  to  construct  the  conic. 

9.  The  tangents  to  a  curve  at  its  infinitely  distant  points 
are  called  its  asymptotes  if  they  pass  through  a  finite  part 
of  the  plane.  Given  the  asymptotes  and  a  finite  point  of  a 
conic,  to  construct  the  conic. 

10.  Given  an  asymptote  and  three  finite  points  on  the 
conic,  to  determine  the  conic. 

11.  Given  four  points,  one  of  which  is  at  infinity,  and 
given  also  that  the  line  at  infinity  is  a  tangent  line)  to 
construct  the  conic. 


CHAPTER  V 

PENCILS  OF  RAYS  OF  THE  SECOND  ORDER 

79.  Pencil  of  rays  of  the  second  order  defined.  If  the 
corresponding  points  of  two  projective  point-rows  be 
joined  by  straight  lines,  a  system  of  lines  is  obtained 
which  is  called  a  pencil  of  rays  of  the  second  order. 
This  name  arises  from  the  fact,  easily  shown  (§  57),  that 
at  most  two  lines  of  the  system  may  pass  through  any 
arbitrary  point  in  the  plane.  For  if  through  any  point 
there  should  pass  three  lines  of  the  system,  then  this 
point  might  be  taken  as  the  center  of  two  projective 
pencils,  one  projecting  one  point-row  and  the  other  pro- 
jecting the  other.  Since,  now,  these  pencils  have  three 
rays  of  one  coincident  with  the  corresponding  rays  of 
the  other,  the  two  are  identical  and  the  two  point-rows 
are  in  perspective  position,  which  was  not  supposed. 

80.  Tangents  to  a  circle.  To  get  a  clear  notion  of  this 
system  of  lines,  we  may  first  show  that  the  tangents 
to  a  circle  form  a  system  of  this  kind.  For  take  any 
two  tangents,  u  and  w',  to  a  circle,  and  let  A  and  B 
be  the  points  of  contact  (Fig.  19).  Let  now  t  be  any 
third  tangent  with  point  of  contact  at  C  and  meeting  u 
and  u'  in  P  and  P'  respectively.  Join  A,  B,  P,  Pf,  and 
C  to  0,  the  center  of  the  circle.  Tangents  from  any 
point  to  a  circle  are  equal,  and  therefore  the  triangles 
POA  and  POC  are  equal,  as  also  are  the  triangles  P'OB 

48 


PENCILS  OF  THE  SECOND  ORDER  49 

and  P'OC.  Therefore  the  angle  POP'  is  constant,  being 
equal  to  half  the  constant  angle  AOC  +COB.  This 
being  true,  if  we  take  any  four  harmonic  points,  Pu  i£, 
JP,  7^,  on  the  line  u,  they  will  project  to  O  in  four 
harmonic  lines,  and  the  tangents 
to  the  circle  from  these  four 
points  will  meet  u'  in  four  har- 
monic points,  i^',  P,',  i^',  i^',  be- 
cause the  lines  from  these  points 
to  0  inclose  the  same  angles  as 
the  lines  from  the  points  Pv  P2, 
J^,  P±  on  u.  The  point-row  on  u  is  therefore  projective 
to  the  point-row  on  u'.'  Thus  the  tangents  to  a  circle 
are  seen  to  join  corresponding  points  on  two  projective 
point-rows,  and  so,  according  to  the  definition,  form  a 
pencil  of  rays  of  the  second  order. 

81.  Tangents  to  a  conic.  If  now  this  figure  be  pro- 
jected to  a  point  outside  the  plane  of  the  circle,  and 
any  section  of  the  resulting  cone  be  made  by  a  plane, 
we  can  easily  see  that  the  system  of  rays  tangent  to  any 
conic  section  is  a  pencil  of  rays  of  the  second  order. 
The  converse  is  also  true,  as  we  shall  see  later,  and  a 
pencil  of  rays  of  the  second  order  is  also  a  set  of  lines 
tangent  to  a  conic  section. 

82.  The  point-rows  u  and  u'  are,  themselves,  lines  of 
the  system,  for  to  the  common  point  of  the  two  point- 
rows,  considered  as  a  point  of  u,  must  correspond  some 
point  of  u',  and  the  line  joining  these  two  corresponding 
points  is  clearly  v!  itself.    Similarly  for  the  line  u. 

83.  Determination  of  the  pencil.  We  now  show  that 
it  is  possible  to  assign  arbitrarily  three  lines,  a,  b,  and  c,  of 


50 


PROJECTIVE  GEOMETRY 


the  system  (besides  the  lines  u  and  w') ;  but  if  these  three 
lines  are  chosen,  the  system  is  completely  determined. 
This  statement  is  equivalent  to  the  following: 

Griven  three  pairs  of  corresponding  points  in  two  pro- 
jective point-rows,  it  is  possible  to  find  a  point  in  one 
which  corresponds  to  any  point  of  the  other. 

We  proceed,  then,  to  the  solution  of  the  fundamental 

Problem.  G-iven  three  pairs  of  points,  AA',  BB\  and 
CC,  of  two  projective  point-rows  u  and  u',  to  find  the  point 
D'  of  u'  which  corresponds  to  any  given  point  D  of  u. 

On  the  line  a,  joining  A  and  A',  take  two  points,  S 
and  S',  as  centers  of  pencils  perspective  to  u  and  u' 
respectively  (Fig.  20).  The  figure 
will  be  much  simplified  if  we  take 
S  on  BB'  and  S'  on  CC.  SA  and 
S'A'  are  corresponding  rays  of  S 
and  S',  and  the  two  pencils  are 
therefore  in  perspective  position. 
It  is  not  difficult  to  see  that  the 
axis  of  perspectivity  m  is  the  line 
joining  B'  and  C.  Given  any  point 
D  on  u,  to  find  the  correspond- 
ing point  D'  on  u'  we  proceed  as 
follows:  Join  D  to  S  and  note 
where  the  joining  line  meets  m.  Join  this  point  to  S!. 
This  last  line  meets  u'  in  the  desired  point  D'. 

We  have  now  in  this  figure  six  lines  of  the  system, 
a,  b,  c,  d,  u,  and  u'.  Fix  now  the  position  of  tf,  u',  b,  c,  and 
d,  and  take  four  lines  of  the  system,  ax,  an,  a3,  a4,  which 
meet  b  in  four  harmonic  points.    These  points  project  to 


Fig.  20 


PENCILS  OF  THE  SECOND  ORDER 


51 


D,  giving  four  harmonic  points  on  m.  These  again  project 
to  Z>',  giving  four  harmonic  points  on  ft  It  is  thus  clear 
that  the  rays  a ,  «2,  a  ,  a4  cut  out  two  projective  point- 
rows  on  any  two  lines  of  the  system.  Thus  u  and  u'  are 
not  special  rays,  and  any  two  rays  of  the  system  will 
serve  as  the  point-rows  to  generate  the  system  of  lines. 

84.  Brianchon's  theorem.  From  the  figure  also  appears 
a  fundamental  theorem  due  to  Brianchon : 

If  1,  2,  3,  4,  5,  G  are  any  six  rays  of  a  pencil  of  the 
second  order,  then  the  lines  Z  =  (12,  45),  m  =  (23,  56), 
w=(34,  61)  all  pass  through  a  point. 

85.  To  make  the  notation  fit  the  figure  (Fig.  21),  make 
0s=l,  b  =  2,  w'=3,  d  =  4,  u  =  5,  6' =  6;  or,  interchanging 
two  of  the  lines,  a=l, 
<?=2,w=3,&=4,«*'c=51 

b  —  {j.  Thus,  by  dif- 
ferent namings  of  the 
lines,  it  appears  that 
not  more  than  GO  dif- 
ferent Brianchon  points 
are  possible.  If  we 
call  12  and  45  oppo- 
site vertices  of  a  cir- 
cumscribed hexagon, 
then  Brianchon's  theorem  may  be  stated  as  follows: 
The  three  lines  joining  the  three  pairs  of  opposite  vertices 
of  a  hexagon  circumscribed  about  a  conic  meet  in  a  point. 

86.  Construction  of  the  pencil  by  Brianchon's  theorem. 
Brianchon's  theorem  furnishes  a  ready  method  of  deter- 
mining a  sixth  line  of  the  pencil  of  rays  of  the  second 


Fig.  21 


52 


PROJECTIVE  GEOMETRY 


Fig.  22 


order  when  five  are  given.  Thus,  select  a  point  in  line 
1  and  suppose  that  line  6  is  to  pass  through  it.  Then 
I  =  (12,  45),  n  =  (34,  61),  and  the  line  m  =  (23,  56)  must 
pass  through  (I,  n).  Then  (23,  In)  meets  5  in  a  point  of 
the  required  sixth  line. 

87.  Point  of  contact 
of  a  tangent  to  a  conic. 
If  the  line  2  approach  as 
a  limiting  position  the 
line  1,  then  the  intersec- 
tion (1,  2)  approaches 
as  a  limiting  position 
the  point  of  contact  of 
1  with  the  conic.  This  suggests  an  easy  way  to  con- 
struct the  point  of  contact  of  any  tangent  with  the  conic. 
Thus  (Fig.  22),  given  the  lines  1,  2,  3,  4,  5  to  construct 
the  point  of  contact  of  1=6. 
Draw  I = (12*  45),  m =(28, 56); 
then  (34,  Im)  meets  1  in  the 
required  point  of  contact  T. 

88.  Circumscribed  quadrilat- 
eral. If  two  pairs  of  lines  in 
Brianchon's  hexagon  coalesce, 
we  have  a  theorem  concern- 
ing a  quadrilateral  circum- 
scribed about  a  conic.  It  is 
easily  found  to  be   (Fig.  23) 

The  four  lines  joining  the  two  opposite  pairs  of  vertices 
and  the  two  opposite  points  of  contact  of  a  quadrilateral 
circumscribed  about  a  conic  all  meet  in  a  point.  The 
consequences  of  this  theorem  will  be  deduced  later. 


PENCILS  OF  THE  SECOND  ORDER  53 

89.  Circumscribed  triangle.  The  hexagon  may  further 
degenerate  into  a  triangle,  giving  the  theorem  (Fig.  24) 

TJie  lines  joining  the  vertices  to 
the  points  of  contact  of  the  opposite 
sides  of  a  triangle  circumscribed 
about  a  conic  all  meet  in  a  point. 

90.  Brianchon's  theorem  may 
also  be  used  to  solve  the  follow- 
ing problems: 

Given  four  tangents  and  the  point 
of  contact  on  any  one  of  them,  to  construct  other  tangents  to 
a  conic.  Given  three  tangents  and  the  points  of  contact  of 
any  two  of  them,  to  construct  oilier  tangents  to  a  conic. 

91.  Harmonic  tangents.  We  have  seen  that  a  variable 
tangent  cuts  out  on  any  two  fixed  tangents  projective 
point-rows.  It  follows  that  if  four  tangents  cut  a  fifth 
in  four  harmonic  points,  they  must  cut  every  tangent  in 
four  harmonic  points.  It  is  possible,  therefore,  to  make 
the  following  definition : 

Four  tangents  to  a  conic  are  said  to  be  harmonic  when 
they  meet  every  other  tangent  in  four  harmonic  points. 

92.  Projectivity  and  perspectivity.  This  definition  sug- 
gests the  possibility  of  defining  a  projective  correspond- 
ence between  the  elements  of  a  pencil  of  rays  of  the 
second  order  and  the  elements  of  any  form  heretofore 
discussed.  In  particular,  the  points  on  a  tangent  are 
said  to  be  perspectively  related  to  the  tangents  of  a  conic 
when  each  point  lies  on  the  tangent  which  corresponds 
to  it.  These  notions  are  of  importance  in  the  higher 
developments  of  the  subject. 


54 


PROJECTIVE  GEOMETRY 


93.  Brianchon's  theorem  may  also  be  applied  to  a 
degenerate  conic  made  up  of  two  points  and  the  lines 
through  them.  Thus(Fig.  25), 

If  a,  b,  c  are  three  lines 
through  a  point  S,  and  a',  b', 
c1  are  three  lines  through  an- 
other point  S\  then  the  fines 
l  =  (ab',  a'b),  m  =  (&</,  b'c~), 
and  n  =  (ca\  c'd)  all  meet  in 
a  point. 

94.  Law  of  duality.  The 
observant  student  will  not 
have  failed  to  note  the  re- 
markable similarity  between  the  theorems  of  this  chap- 
ter and  those  of  the  preceding.  He  will  have  noted 
that  points  have  replaced  lines  and  lines  have  replaced 
points ;  that  points  on  a  curve  have  been  replaced  1  >y 
tangents  to  a  curve ;  that  pencils  have  been  replaced 
by  point-rows,  and  that  a  conic  considered  as  made  up 
of  a  succession  of  points  has  been  replaced  by  a  conic 
considered  as  generated  by  a  moving  tangent  line.  The 
theory  upon  which  this  wonderful  laiv  of  duality  is  based 
will  be  developed  in  the  next  chapter. 


Fig.  25 


PROBLEMS 

1.  Given  four  lines  in  the  plane,  to  construct  another 
which  shall  meet  them  in  four  harmonic  points. 

2.  Where  are  all  such  lines  found? 

3.  Given  any  five  lines  in  the  plane,  construct  on  each 
the  point  of  contact  with  the  conic  tangent  to  them  all. 


PENCILS  OF  THE  SECOND  ORDER  55 

4.  Given  four  lines  and  the  point  of  contact  on  one,  to 
construct  the  conic.  ("  To  construct  the  conic  "  means  here 
to  draw  as  many  other  tangents  as  may  be  desired.) 

5.  Given  three  lines  and  the  point  of  contact  on  two  of 
them,  to  construct  the  conic. 

6.  Given  four  lines  and  the  line  at  infinity,  to  construct 
the  conic. 

7.  Given  three  lines  and  the  line  at  infinity,  together 
with  the  point  of  contact  at  infinity,  to  construct  the  conic. 

8.  Given  three  lines,  two  of  which  are  asymptotes,  to 
construct  the  conic. 

9.  Given  five  tangents  to  a  conic,  to  draw  a  tangent 
which  shall  be  parallel  to  any  one  of  them. 

10.  The  lines  a,  b,  c  are  drawn  parallel  to  each  other. 
The  lines  a',  b',  c'  are  also  drawn  parallel  to  each  other. 
Show  why  the  lines  (ab'}  a'b),  (be',  b'c),  (ca'}  c'a)  meet  in  a 
point.  (In  problems  6  to  10  inclusive,  parallel  lines  are  to 
be  drawn.) 


CHAPTER  VI 


POLES  AND  POLARS 

95.  Inscribed  and  circumscribed  quadrilaterals.  The 
following  theorems  have  been  noted  as  special  cases  of 
Pascal's  and  Brianchon's  theorems: 

If  a  quadrilateral  be  inscribed  in  a  conic,  two  pairs  of 
opposite  sides  and  the  tangents  at  opposite  vertices  inter- 
sect in  four  points,  all  of  which  lie  on  a  straight  line. 

If  a  quadrilateral  be  circumscribed  about  a  conic,  the 
lines  joining  two  pairs  of  opposite  vertices  and  the  linet 
joining  two  opposite  points  of  contact  are  four  lines  which 
meet  in  a  point. 

96.  Definition  of  the  polar  line  of  a  point.  Consider 
the  quadrilateral  K,  L,  31,  N  inscribed  in  the  conic 
(Fig.  26).     It 


determines  the 
four  harmonic 
points  A,  B,  G, 
D  which  pro- 
ject from  N  in  to 
the  four  har- 
monic points  M, 
B,  K,  0.  Now 
the  tangents  at  K  and  M  meet  in  P,  a  point  on  the 
line  AB.    The  line  AB  is  thus  determined  entirely  by 

56 


Fig.  26 


POLES  AND  POLAKS  57 

the  point  0.  For  if  we  draw  any  line  through  it,  meeting 
the  conic  in  K  and  M,  and  construct  the  harmonic 
conjugate  B  of  0  with  respect  to  K  and  M,  and  also 
the  two  tangents  at  K  and  M  which  meet  in  the  point 
P,  then  BP  is  the  line  in  question.  It  thus  appears 
that  the  line  LON  may  be  any  line  whatever  through  0 ; 
and  since  D,  L,  0,  A7"  are  four  harmonic  points,  we 
may  describe  the  line  AB  as  the  locus  of  points  which 
are  harmonic  conjugates  of  0  with  respect  to  the  two 
points  where  any  line  through  0  meets  the  curve. 

97.  Furthermore,  since  the  tangents  at  L  and  N  meet 
on  this  same  line,  it  appears  as  the  locus  of  intersections 
of  pairs  of  tangents  drawn  at  the  extremities  of  chords 
through  0. 

98.  This  important  line,  which  is  completely  deter- 
mined by  the  point  0,  is  called  the  polar  of  0  with 
respect  to  the  conic ;  and  the  point  0  is  called  the  pole 
of  the  line  with  respect  to  the  conic. 

99.  If  a  point  B  is  on  the  polar  of  0,  then  it  is  har- 
monically conjugate  to  0  with  respect  to  the  two  inter- 
sections K  and  M  of  the  line  BO  with  the  conic.  But 
for  the  same  reason  0  is  on  the  polar  of  B.  We  have, 
then,  the  fundamental  theorem 

If  one  point  lies  on  the  polar  of  a  second,  then  the 
second  lies  on  the  polar  of  the  first. 

100.  Conjugate  points  and  lines.  Such  a  pair  of  points 
are  said  to  be  conjugate  with  respect  to  the  conic.  Simi- 
larly, lines  are  said  to  be  conjugate  to  each  other  with 
respect  to  the  conic  if  one,  and  consequently  each, 
passes  through  the  pole  of  the  other. 


58  PROJECTIVE  GEOMETRY 

101.  Construction  of  the  polar  line  of  a  given  point. 
Given  a  point  P,  if  it  is  within  the  conic  (that  is,  if  no 
tangents  may  be  drawn  from  P  to  the 
conic),  we  may  construct  its  polar  line 
by  drawing  through  it  any  two  chords 
and  joining  the  two  points  of  inter- 
section of  the  two  pairs  of  tangents 
at  their  extremities.    If  the  point  P  is 

outside  the  conic,  we  may  draw  the  two  tangents  and 
construct  the  chord  of  contact  (Fig.  27). 

102.  Self-polar  triangle.  In  Fig.  26  it  is  not  difficult 
to  see  that  AOC  is  a  self-polar  triangle,  that  is,  cadi 
vertex  is  the  pole  of  the  opposite  side.  For  B,  M,  O,  K 
are  four  harmonic  points,  and  they  project  to  C  in  four 
harmonic  rays.  The  line  CO,  therefore,  meets  the  line 
AMN  in  a  point  on  the  polar  of  A,  being  separated  from 
A  harmonically  by  the  points  M  and  N.  Similarly,  the 
line  CO  meets  KL  in  a  point  on  the  polar  of  A,  and 
therefore  CO  is  the  polar  of  A.  Similarly,  OA  is  the 
polar  of  C,  and  therefore  0  is  the  pole  of  AC. 

103.  Pole  and  polar  projectively  related.  Another  very 
important  theorem  comes  directly  from  Fig.  26. 

As  a  point  A  moves  along  a  straight  line  its  polar  with 
respect  to  a  conic  revolves  about  a  fixed  point  arid  describe* 
a  pencil  projective  to  the  point-row  described  by  A. 

For,  fix  the  points  L  and  N  and  let  the  point  A  move 
along  the  line  AQ;  then  the  point-row  A  is  projective 
to  the  pencil  LK,  and  since  K  moves  along  the  conic, 
the  pencil  LK  is  projective  to  the  pencil  NK,  which  in 
turn  is  projective  to  the  point-row  C,  which,  finally,  is 
projective  to  the  pencil  OC,  which  is  the  polar  of  A. 


POLES  AND  POLARS  59 

104.  Duality.  We  have,  then,  in  the  pole  and  polar 
relation  a  device  for  setting  up  a  one-to-one  correspond- 
ence between  the  points  and  lines  of  the  plane  —  a  cor- 
respondence which  may  be  called  projective,  because  to 
four  harmonic  points  or  lines  correspond  always  four 
harmonic  lines  or  points.  To  every  figure  made  up  of 
points  and  lines  will  correspond  a  figure  made  up  of 
lines  and  points.  To  a  point-row  of  the  second  order, 
which  is  a  conic  considered  as  a  point-locus,  corresponds 
a  pencil  of  rays  of  the  second  order,  which  is  a  conic 
considered  as  a  line-locus.  The  name  {  duality '  is  used 
to  describe  this  sort  of  correspondence.  It  is  important 
to  note  that  the  dual  relation  is  subject  to  the  same 
exceptions  as  the  one-to-one  correspondence  is,  and 
must  not  be  appealed  to  in  cases  where  the  one-to-one 
correspondence  breaks  down.  We  have  seen  that  there 
is  in  Euclidean  geometry  one  and  only  one  ray  in  a 
pencil  which  has  no  point  in  a  point-row  perspective  to 
it  for  a  corresponding  point ;  namely,  the  line  parallel 
to  the  line  of  the  point-row.  Any  theorem,  therefore, 
that  involves  explicitly  the  point  at  infinity  is  not  to 
be  translated  into  a  theorem  concerning  lines.  Further, 
in  the  pencil  the  angle  between  two  lines  has  nothing 
to  correspond  to  it  in  a  point-row  perspective  to  the 
pencil.  Any  theorem,  therefore,  that  mentions  angles  is 
not  translatable  into  another  theorem  by  means  of  the 
law  of  duality.  Now  we  have  seen  that  the  notion  of 
the  infinitely  distant  point  on  a  line  involves  the  notion 
of  dividing  a  segment  into  any  number  of  equal  parts  — 
in  other  words,  of  measuring.  If,  therefore,  we  call  any 
theorem  that  has  to  do  with  the  line  at  infinity  or  with 


60  PROJECTIVE  GEOMETRY 

the  measurement  of  angles  a  metrical  theorem,  and  any 
other  kind  a  projective  theorem,  we  may  put  the  case 
as  follows: 

Any  projective  theorem  involves  another  theorem,  dual  to 
if,  obtainable  by  interchanging  everywhere  the  words  ''point'' 
and  "line? 

105.  Self -dual  theorems.  The  theorems  of  this  chap- 
ter will  be  found,  upon  examination,  to  be  self -dual; 
that  is,  no  new  theorem  results  from  applying  the 
process  indicated  in  the  preceding  paragraph.  It  is 
therefore  useless  to  look  for  new  results  from  the  theo- 
rem on  the  circumscribed  quadrilateral  derived  from 
Brianchon's,  which  is  itself  clearly  the  dual  of  Pascal's 
theorem,  and  in  fact  was  first  discovered  by  dualization 
of  Pascal's. 

106.  It  should  not  be  inferred  from  the  above  discus- 
sion that  one-to-one  correspondences  may  not  be  devised 
that  will  control  certain  of  the  so-called  metrical  rela- 
tions. A  very  important  one  may  be  easily  found  that 
leaves  angles  unaltered.  The  relation  called  similarity 
leaves  ratios  between  corresponding  segments  unaltered. 
The  above  statements  apply  only  to  the  particular  one- 
to-one  correspondence  considered. 

PROBLEMS 

1.  Given  a  quadrilateral,  construct  the  quadrangle  polar 
to  it  with  respect  to  a  given  conic. 

2.  A  point  moves  along  a  straight  line.  Show  that  its 
polar  lines  with  respect  to  two  given  conies  generate  a 
point-row  of  the  second  order. 


POLES  AND  POLARS  61 

3.  Given  five  points,  draw  the  polar  of  a  point  with  re- 
spect to  the  conic  passing  through  them,  without  drawing 
the  conic  itself. 

4.  Given  five  lines,  draw  the  polar  of  a  point  with  re- 
spect to  the  conic  tangent  to  them,  without  drawing  the 
conic  itself. 

5.  Dualize  problems  3  and  4. 

6.  Given  four  points  on  the  conic,  and  the  tangent  at  one 
of  them,  draw  the  polar  of  a  given  point  without  drawing 
the  conic.    Dualize. 

7.  A  point  moves  on  a  conic.  Show  that  its  polar  line 
with  respect  to  another  conic  describes  a  pencil  of  rays  of 
the  second  order. 

Suggestion.   Replace  the  given  conic  "by  a  pair  of  projective  pencils. 

8.  Show  that  the  poles  of  the  tangents  of  one  conic  with 
respect  to  another  lie  on  a  conic. 

9.  The  polar  of  a  point  A  with  respect  to  one  conic  is  a, 
and  the  pole  of  a  with  respect  to  another  conic  is  A '.  Show 
that  as  A  travels  along  a  line,  A '  also  travels  along  another 
line.  In  general,  if  A  describes  a  curve  of  degree  n,  show 
that  A '  describes  another  curve  of  the  same  degree  n.  (The 
degree  of  a  curve  is  the  greatest  number  of  points  that  it 
may  have  in  common  with  any  line  in  the  plane.) 


CHAPTER  VII 

METRICAL  PROPERTIES  OF  THE  CONIC  SECTIONS 

107.  Diameters.  Center.  After  what  has  been  said  in 
the  last  chapter  one  would  naturally  expect  to  get  at 
the  metrical  properties  of  the  conic  sections  by  the 
introduction  of  the  infinite  elements  in  the  plane.  En- 
tering into  the  theory  of  poles  and  polars  with  these 
elements,  we  have  the  following  definitions : 

The  polar  line  of  an  infinitely  distant  point  is  called 
a  diameter,  and  the  pole  of  the  infinitely  distant  line  is 
called  the  center,  of  the  conic. 

108.  From  the  harmonic  properties  of  poles  and  polars, 
The  center  bisects  all  chords  through  it  (§  39). 
Every  diameter  passes  through  the  center. 

All  chords  through  the  same  point  at  infinity  (that  in, 
each  of  a  set  of  parallel  chords^)  are  bisected  by  the  diameter 
which  is  the  polar  of  that  infinitely  distant  point. 

109.  Conjugate  diameters.  We  have  already  denned 
conjugate  lines  as  lines  which  pass  each  through  the 
pole  of  the  other  (§  100). 

Any  diameter  bisects  all  chords  parallel  to  its  conjugate. 

The  tangents  at  the  extremities  of  any  diameter  are 
parallel,  and  parallel  to  the  conjugate  diameter. 

Diameters  parallel  to  the  sides  of  an  inscribed  paral- 
lelogram are  conjugate. 

All  these  theorems  are  easy  exercises  for  the  student. 


METRICAL  PROPERTIES  63 

110.  Classification  of  conies.  Conies  are  classified  ac- 
cording to  their  relation  to  the  infinitely  distant  line. 
If  a  conic  has  two  points  in  common  with  the  line  at 
infinity,  it  is  called  a  hyperbola ;  if  it  has  no  point  in 
common  with  the  infinitely  distant  line,  it  is  called  an 
ellipse ;  if  it  is  tangent  to  the  line  at  infinity,  it  is  called 
a  parabola. 

111.  In  a  hyperbola  the  center  is  outside  the  curve 
(§  101),  since  the  two  tangents  to  the  curve  at  the  points 
where  it  meets  the  line  at  infinity  determine  by  their 
intersection  the  center.  As  previously  noted,  these  two 
tangents  are  called  the  asymptotes  of  the  curve.  The 
ellipse  and  the  parabola  have  no  asymptotes. 

112.  The  center  of  the  parabola  is  at  infinity,  and  there- 
fore all  its  diameters  are  parallel,  for  the  pole  of  a  tan- 
gent line  is  the  point  of  contact. 

The  locus  of  the  middle  points  of  a  series  of  parallel 
chords  in  a  parabola  is  a  diameter,  and  the  direction  of 
the  line  of  eenters  is  the  same  for  all  series  of  parallel 
chords. 

The  center  of  an  ellipse  is  within  the  curve. 

113.  Theorems  concerning  asymptotes.  We  derived  as 
a  consequence  of  the  theorem  of  Brianchon  (§  89)  the 
proposition  that  if  a  triangle  be  circumscribed  about 
a  conic,  the  lines  joining  the  vertices  to  the  points 
of  contact  of  the  opposite  sides  all  meet  in  a  point. 
Take,  now,  for  two  of  the  tangents  the  asymptotes  of 
a  hyperbola,  and  let  any  third  tangent  cut  them  in  A 
and  B  (Fig.  28).  If,  then,  0  is  the  intersection  of  the 
asymptotes,  —  and  therefore  the  center  of  the  curve,  — 


64 


PROJECTIVE  GEOMETRY 


Fig.  28 


then  the  triangle  OAB  is  circumscribed  about  the  curve. 
By  the  theorem  just  quoted,  the  line  through  A  par- 
allel to  OB,  the  line  through  B  parallel  to  OA,  and  the 
line  OP  through  the  point  of 
contact  of  the  tangent  AB 
all  meet  in  a  point  C.  But 
OACB  is  a  parallelogram,  and 
PA  =  PB.    Therefore 

The  asymptotes  cut  off  on 
each  tangent  a  segment  which  is 
bisected  by  the  point  of  contact. 

114.  If  we  draw  a  line  OQ 
parallel  to  AB,  then  OP  and  OQ  are  conjugate  diam- 
eters, since  OQ  is  parallel  to  the  tangent  at  the  point 
where  OP  meets  the  curve.  Then,  since  A,  P,  B,  and 
the  point  at  infinity  on  AB  are  four  harmonic  points, 
we  have  the  theorem 

Conjugate  diameters  of  the  hyperbola  are  harmonic 
conjugates  with   respect    to    the    asymptotes. 

115.  The  chord  A"B",  parallel  to  the  diameter  OQ,  is 
bisected  at  P'  by  the  conjugate  diameter  OP.  If  the 
chord  A"B"  meet  the  asymptotes  in  A',  B',  then  A',  P',  B', 
and  the  point  at  infinity  are  four  harmonic  points,  and 
therefore  P'  is  the  middle  point  of  A'B'.  Therefore 
A'A"=B'B"  and  we  have  the  theorem 

Tlte  segments  cut  off  on  any  chord  between  the  hyperbola 
and  its  asymptotes  are  equal. 

116.  This  theorem  furnishes  a  ready  means  of  con- 
structing the  hyperbola  by  points  when  a  point  on  the 
curve  and  the  two  asymptotes  are  given. 


METRICAL  PROPERTIES 


65 


Fig.  29 


117.  For  the  circumscribed  quadrilateral,  Brianchon's 
theorem  gave  (§  88)  The  lines  joining  opposite  vertices 
and  the  lines  joining  opposite  points  of  contact  are  four 
lines  meeting  in  a  point.  Take  now  for  two  of  the 
tangents  the  asymptotes,  and  let  AB  and  CD  be  any 
other  two  (Fig.  29). 
If  B  and  D  are  op- 
posite vertices,  and 
also  A  and  C,  then 
A  C  and  BD  are  par- 
allel, and  parallel  to 
FQ,  the  line  joining 
the  points  of  con- 
tact of  AB  and  CD, 
for  these  are  three  of 
the  four  lines  of  the 
theorem  just  quoted.  The  fourth  is  the  line  at  infinity 
which  joins  the  point  of  contact  of  the  asymptotes.  It 
is  thus  seen  that  the  triangles  ABC  and  ADC  are 
equivalent,  and  therefore  the  triangles  AOB  and  COD 
are  also.  The  tangent  AB  may  be  fixed,  and  the  tangent 
CD  chosen  arbitrarily;  therefore 

The  triangle  formed  by  any  tangent  to  the  hyperbola 
and  the  two  asymptotes  is  of  constant  area. 

118.  Equation  of  hyperbola  referred  to  the  asymptotes. 
Draw  through  the  point  of  contact  P  of  the  tangent 
AB  two  lines,  one  parallel  to  one  asymptote  and  the 
other  parallel  to  the  other.  One  of  these  lines  meets 
OB  at  a  distance  y  from  O,  and  the  other  meets  OA  at 
a  distance  x  from  0.    Then,  since  P  is  the  middle  point 


66 


PROJECTIVE  GEOMETRY 


of  AB,  x  is  one  half  of  OA  and  y  is  one  half  of  OB. 
The  area  of  the  parallelogram  whose  adjacent  sides  arc 
x  and  y  is  one  half  the  area  of  the  triangle  AOB,  and 
therefore,  by  the  preceding  paragraph,  is  constant.  This 
area  is  equal  to  xy  •  sin  a,  where  a  is  the  constant  angle 
between  the  asymptotes.  It  follows  that  the  product  ."/ 
is  constant,  and  since  x  and  y  are  the  oblique  coordi- 
nates of  the  point  P,  the  asymptotes  being  the  axes 
of  reference,  we  have 

The  equation  of  the  hyperbola,  referred  to  the  asymptote* 
as  axes,  is  xy  =  constant. 

This  identifies  the  curve  with  the  hyperbola  as  de- 
fined  and  discussed    in  works    on    analytic    geometry. 

119.  Equation  of 
parabola.  We  have 
defined  the  parabola 
as  a  conic  which  is 
tangent  to  the  line 
at  infinity  (§  110). 
Draw  now  two  tan- 
gents to  the  curve 
(Fig.  30),  meeting  in 
A,  the  points  of  con- 
tact being  B  and  C. 
These  two  tangents, 
together  with  the 
line  at  infinity,  form 
a  triangle  circum- 
scribed    about     the 

conic.  Draw  through  B  a  parallel  to  AC,  and  through 
C  a  parallel  to  AB.    If  these  meet  in  D,  then  A  l>  is  a 


Fig.  30 


METRICAL  PROPERTIES  67 

diameter.  Let  AD  meet  the  curve  in  P,  and  the  chord 
BC  in  Q.  P  is  then  the  middle  point  of  AQ.  Also,  Q 
is  the  middle  point  of  the  chord  BCS  and  therefore  the 
diameter  AD  bisects  all  chords  parallel  to  BC.  In  par- 
ticular, AD  passes  through  P,  the  point  of  contact  of 
the  tangent  drawn  parallel  to  BC. 

Draw  now  another  tangent,  meeting  AB  in  B'  and  AC 
in  C'.  Then  these  three,  with  the  line  at  infinity,  make 
a  circumscribed  quadrilateral.  But,  by  Brianchon's  the- 
orem applied  to  a  quadrilateral  (§  88),  it  appears  that  a 
parallel  to  AC  through  B',  a  parallel  to  AB  through  C', 
and  the  line  BC  meet  in  a  point  D'.  Also,  from  the  similar 
triangles  BB'D'  and  BA  C  we  have,  for  all  positions  of  the 
tangent  lineP'C",  ## .  mr==AC:  AB, 

or,  since  B'D'  =  AC, 

AC :  BB'  =AC:AB  =  constant. 

If  another  tangent  meet  AB  in  B"  and  A  C  in  C",  we  have 

AC  :  BB1  =  AC" :  BB", 

and  by  subtraction  we  get 

CC":B'B"  =  constant; 
whence 

The  segments  cut  off  on  any  two  tangents  to  a  parabola 
by  a  variable  tangent  are  proportional. 

If  now  we  take  the  tangent  B'C  as  axis  of  ordinates, 
and  the  diameter  through  the  point  of  contact  0  as  axis 
of  abscissas,  calling  the  coordinates  of  B  (x,  y)  and  of 
C  (V,  y'~),  then,  from  the  similar  triangles  BMD'  and 
CQ'D',  we  have 

y:y'=  BD'  :D'C=  BB' :  AB'. 
Also  y :  y'  =  B'D' :C'C=AC:  C'C. 


68  PROJECTIVE  GEOMETRY 

If  now  a  line  is  drawn  through  A  parallel  to  a  diameter, 
meeting  the  axis  of  ordinates  in  K,  we  have 
AK  :OQ'  =  AC':CC'  =  y:  y', 
and  OM:AK=BB':  AB'  =  y:y', 

and,  by  multiplication, 

OM:OQ'  =  y*:y'\ 
or  x :  x!  =  y'2 :  y'2 ; 

whence 

The  abscissas  of  two  points  on  a  parabola  are  to  each 
other  as  the  squares  of  the  corresponding  coordinates,  a 
diameter  and  the  tangent  to  the  curve  at  the  extremity  vf 
the  diameter  being  the  axes  of  reference. 

The  last  equation  may  be  written 
y2  =  2px, 
where  2p  stands  for  y'2:x'. 

The  parabola  is  thus  identified  with  the  curve  of  the 
same  name  studied  in  treatises  on  analytic  geometry. 

120.  Equation  of  central  conies  referred  to  conjugate 
diameters.  Consider  now  a  central  conic,  that  is,  one 
which  is  not  a  parabola  and  the  center  of  which  is 
therefore  at  a  finite  distance.  Draw  any  four  tangents 
to  it,  two  of  which  are  parallel  (Fig.  31).  Let  the 
parallel  tangents  meet  one  of  the  other  tangents  in  A 
and  B  and  the  other  in  C  and  D,  and  let  P  and  Q  be 
the  points  of  contact  of  the  parallel  tangents  R  and  S 
of  the  others.  Then  AC,  BD,  PQ,  and  ^*S'  all  meet  in 
a  point  W  (§  88).    From  the  figure, 

J 'IV:  WQ  =  AP:QC  =  PD:BQ, 
or  APBQ  =  PD-  QC. 


METRICAL  PROPERTIES 


69 


If  now  DC  is  a  fixed  tangent  and  AB  a  variable  one, 
we  have  from  this  equation 

AP  •  BQ  =  constant. 

This  constant  will  be  positive  or  negative  according  as 
PA  and  BQ  are  measured  in  the  same  or  in  opposite 
directions.    Accordingly  we  write 

AP  •  BQ  =  ±  b\ 

Since  AD  and  BC  are  parallel  tangents,  PQ  is  a  diam- 
eter and  the  conjugate  diameter  is  parallel  to  AD.  The 
middle  point  of  PQ  is  the 
center  of  the  conic.  We  take 
now  for  the  axis  of  abscissas 
the  diameter  PQ,  and  the 
conjugate  diameter  for  the 
axis  of  ordinates.  Join  A  to 
Q  and  B  to  P  and  draw  a 
line  through  S  parallel  to 
the  axis  of  ordinates.  These 
three  lines  all  meet  in  a  point 
N,  because  AP,  BQ,  and  AB 
form  a  triangle  circumscribed 
to  the  conic.     Let  NS  meet 

PQ  in  M.  Then,  from  the  properties  of  the  circum- 
scribed triangle  (§  89),  M,  N,  S,  and  the  point  at  infinity 
on  NS  are  four  harmonic  points,  and  therefore  N  is  the 
middle  point  of  MS.  If  the  coordinates  of  S  are  (x,  y), 
so  that  OM  is  x  and  MS  is  y,  then  MN=  y/1.  Now 
from  the  similar  triangles  PMN  and  PQB  we  have 

BQ:PQ  =  NM.PM, 


Fig.  31 


70  PROJECTIVE  GEOMETRY 

and  from  the  similar  triangles  PQA  and  MQN, 

AP:PQ  =  MN:MQ, 
whence,  multiplying,  we  have 

±  62/4  a2  =  y2/4  (a  +  x)  (a  -  sr> 

where  a  =  -~ , 

or,  simplifying,  x2/a2  +  y~/±  b2  =  1, 
which  is  the  equation  of  an  ellipse  when  b2  has  a  posi- 
tive sign,  and  of  a  hyperbola  when  b2  has  a  negative 
sign.  We  have  thus  identified  point-rows  of  the  second 
order  with  the  curves  given  by  equations  of  the  second 
degree. 

PROBLEMS 

1.  Draw  a  chord  of  a  given  conic  which  shall  be  bisected 
by  a  given  point  P. 

2.  Show  that  all  chords  of  a  given  conic  that  are  bisected 
by  a  given  chord  are  tangent  to  a  parabola. 

3.  Construct  a  parabola,  given  two  tangents  with  their 
points  of  contact. 

4.  Construct  a  parabola,  given  three  points  and  the  direc- 
tion of  the  diameters. 

5.  A  line  u'  is  drawn  through  the  pole  IT  of  a  line  u  and 
at  right  angles  to  u.  The  line  u  revolves  about  a  point  P. 
Show  that  the  line  u'  is  tangent  to  a  parabola.  (The  lines  u 
and  u'  are  called  normal  conjugates.) 

6.  Given  a  conic  and  its  center  O,  to  draw  a  line  through 
a  given  point  P  parallel  to  a  given  line  q.  Prove  the  fol- 
lowing construction :  Let  p  be  the  polar  of  P,  Q  the  pole  of 
q,  and  A  the  intersection  of  p  with  OQ.  The  polar  of  A  is 
the  desired  line. 


CHAPTER  VIII 


INVOLUTION 


121.  Fundamental  theorem.  The  important  theorem 
concerning  two  complete  quadrangles  (§  26),  upon  which 
the  theory  of  four  harmonic  points  was  based,  can  easily 
be  extended  to 
the  case  where 
the  four  lines 
KL,  K'L',  MN, 
M'N'  do  not 
all  meet  in  the 
same  point  A, 
and  the  more 
general  theo- 
rem that  re- 
sults may  also 

be  made  the  basis  of  a  theory  no  less  important,  which  has 
to  do  with  six  points  on  a  line.  The  theorem  is  as  follows : 

Given  two  complete  quadrangles,  K,  L,  M,  N  and 
K',  L',  M',  N',  so  related  that  KL  and  K'L'  meet  in  A, 
MN  and  M'N'  in  A',  KN  and  K'N'  in  B,  LM  and  L'M 
in  B',  LN  and  L'N'  in  C,  and  KM  and  K'M'  in  C,  then, 
if  A,  A',  B,  B',  and  C  are  in  a  straight  line,  the  point  C 
also  lies  on  that  straight  line. 

The  theorem  follows  from  Desargues's  theorem 
(Fig.  32).     It  is  seen   that  KK',  LL',  MM',  NN'   all 

71 


Fig.  32 


72  PROJECTIVE  GEOMETRY 

meet  in  a  point,  and  thus,  from  the  same  theorem,  ap- 
plied to  the  triangles  KLM  and  K'L'M',  the  point  C'  is  on 
the  same  line  with  A  and  />'.  As  in  the  simpler  case,  it 
is  seen  that  there  is  an  indefinite  number  of  quadrangles 
which  may  be  drawn,  two  sides  of  which  go  through 
A  and  A',  two  through  B  and  7>',  and  one  through  C. 
The  sixth  side  must  then  go  through  C.    Therefore, 

122.  Tivo  pairs  of  points,  A,  A'  and  B,  B',  being  given, 
then  the  point  C'  corresponding  to  any  given  point  C  is 
uniquely  determined. 

The  construction  of  this  sixth  point  is  easily  accom- 
plished. Draw  through  A  and  A'  any  two  lines,  and 
cut  across  them  by  any  line  through  C  in  the  points 
L  and  N.  Join  N  to  B  and  L  to  B',  thus  determining 
the  points  K  and  M  on  the  two  lines  through  A  and  A'. 
The  line  KM  determines  the  desired  point  C'.  Manifestly, 
starting  from  C",  we  come  in  this  way  always  to  the 
same  point  C.  The  particular  quadrangle  employed  is 
of  no  consequence.  Moreover,  since  one  pair  of  opposite 
sides  in  a  complete  quadrangle  is  not  distinguishable 
in  any  way  from  any  other,  the  same  set  of  six  points 
will  be  obtained  by  starting  from  the  pairs  AA'  and 
CC',  or  from  the  pairs  BB'  and  CC'.  . 

123.  Definition  of  involution  of  points  on  a  line. 

Tliree  pairs  of  points  on  a  line  are  said  to  be  in  I » volu- 
tion if  through  each  pair  may  be  drawn  a  pair  of  opposite 
sides  of  a  complete  quadrangle.  If  two  pairs  are  fixed  and 
one  of  the  third  pair  describes  the  line,  then  the  other  also 
describes  the  line,  and  the  points  of  the  line  are  said  to  be 
paired  in  the  involution  determined  by  the  two  fixed  pairs. 


INVOLUTION 


73 


124.  Double-points  in  an  involution.  The  points  C  and 
C  describe  projective  point-rows,  as  may  be  seen  by  fixing 
the  points  L  and  M.  The  self-corresponding  points,  of 
which  there  are  two  or  none,  are  called  the  double-points  in 
the  involution.  It  is  not  difficult  to  see  that  the  double- 
points  in  the  involution  are  harmonic  conjugates  with 
respect  to  corresponding  points  in  the  involution.  For, 
fixing  as  before  the  points  L  and  M,  let  the  intersection 
of  the  lines  CL  and  C'Mbe  P  (Fig.  33).  The  locus  of  P  is 
a  conic  which  goes  through  the  double-points,  because  the 
point-rows  C  and 
C'  are  projective, 
and  therefore  so 
are  the  pencils 
LC  and  MC' 
which  generate 
the  locus  of  P. 
Also,  when  C 
and  C  fall  to- 
gether, the  point 
P  coincides  with 
them.  Further,  the  tangents  at  L  and  M  to  this  conic 
described  by  P  are  the  lines  LB  and  MB.  For  in  the 
pencil  at  L  the  ray  LM  common  to  the  two  pencils  which 
generate  the  conic  is  the  ray  LB'  and  corresponds  to  the 
ray  MB  of  If,  which  is  therefore  the  tangent  line  to  the 
conic  at  M.  Similarly  for  the  tangent  LB  at  L.  LM  is 
therefore  the  polar  of  B  with  respect  to  this  conic,  and 
B  and  B'  are  therefore  harmonic  conjugates  with  respect 
to  the  double-points.  The  same  discussion  applies  to  any 
other  pair  of  corresponding  points  in  the  involution. 


Fig.  33 


74 


PROJECTIVE  GEOMETRY 


125.  Desargues's  theorem  concerning  conies  through 
four  points.  Let  DD'  be  any  pair  of  points  in  the  in- 
volution determined  as  above,  and  consider  the  conic 
passing  through  the  five  points  A',  A,  M,  N,  I>.  We 
shall  use  Pascal's  theorem  to  show  that  this  conic  also 
passes  through  D'.  The  point  D'  is  determined  as  fol- 
lows :  Fix  L  and  M  as  before  (Fig.  34)  and  join  D  to  L, 
giving  on  MN 
the  point  N'. 
Join  N'  to  7?, 
giving  on  LK 
the  point  K'. 
Then  MK'  de- 
termines the 
point  D'  on 
the  line  A .  I ', 
given  by  the 
complete  quad- 
rangle K',  L,  3A,  N'.  Consider  the  following  six  points, 
numbering  them  in  order:  D=l,  D'=  2,  J/=3,  7V=  4, 
X=5,  and  L  =  Q.  We  have  the  following  intersections: 
A?  =(12-45),  AC'=  (23-56),  N'=  (34-61);  and  since  by 
construction  B,  JV',  and  K'  are  on  a  straight  line,  it  fol- 
lows from  the  converse  of  Pascal's  theorem,  which  is 
easily  established,  that  the  six  points  are  on  a  conic. 
We  have,  then,  the  beautiful  theorem  due  to  Desargues : 

The  system  of  conies  through  four  points  meets  any  line 
in  the  plane  in  pairs  of  points  in  involution. 

126.  It  appears  also  that  the  six  points  in  involution 
determined  by  the  quadrangle  through  the  four  fixed 


Fig.  34 


INVOLUTION 


75 


points  belong  also  to  the  same  involution  with  the 
points  cut  out  by  the  system  of  conies,  as  indeed  we 
might  infer  from  the  fact  that  the  three  pairs  of  oppo- 
site sides  of  the  quadrangle  may  be  considered  as 
degenerate  conies  of  the  system. 

127.  Conies  through  four  points  touching  a  given  line. 
It  is  further  evident  that  the  involution  determined  on 
a  line  by  the  system  of  conies  will  have  a  double-point 
where  a  conic  of  the  system  is  tangent  to  the  line.  We 
may  therefore  infer  the  theorem 

Tlirough  four  fixed  points  in  the  ^Zawe  tico  conies  or 
none  may  he  drawn  tangent  to  any  given  line. 

128.  Double  correspondence.  We  have  seen  that  cor- 
responding points  in  an  involution  form  two  projective 
point-rows  superposed  on  the  same  straight  line.  Two 
projective  point-rows  superposed 
on  the  same  straight  line  are,  how- 
ever, not  necessarily  in  involution, 
as  a  simple  example  will  show. 
Take  two  lines,  a  and  a',  which 
both  revolve  about  a  fixed  point  S 
and  which  always  make  the  same 
angle  with  each  other  (Fig.  35). 
These  lines  cut  out  on  any  line 
in  the  plane  which  does  not  pass 
through  S  two  projective  point- 
rows,  which  are  not,  however,  in 

involution  unless  the  angle  between  the  lines  is  a  right 
angle.  For  a  point  P  may  correspond  to  a  point  P', 
which    in    turn  will    correspond  to    some    other   point 


Fig.  35 


76  PROJECTIVE  GEOMETRY 

than  P.  The  peculiarity  of  point-rows  in  involution 
is  that  any  point  will  correspond  to  the  same  point, 
in  whichever  point-row  it  is  considered  as  belonging. 
In  this  case,  if  a  point  P  corresponds  to  a  point  P\  then 
the  point  P'  corresponds  back  again  to  the  point  P. 
The  points  P  and  P'  are  then  said  to  correspond  doubly. 
This  notion  is  worthy  of  further  study. 

129.  Steiner's  construction.  It  will  be  observed  that 
the  solution  of  the  fundamental  problem  given  in  §  83, 
Given  three  pairs  of  points  of  two  projective  point-rows,  to 


construct  other  pairs,  cannot  be  carried  out  if  the  two 
point-rows  lie  on  the  same  straight  line.  Of  course  the 
method  may  be  easily  altered  to  cover  that  case  also, 
but  it  is  worth  while  to  give  another  solution  of  the 
problem,  due  to  Steiner,  which  will  also  give  further 
information  regarding  the  theory  of  involution,  and 
which  may,  indeed,  be  used  as  a  foundation  for  that 
theory.  Let  the  two  point-rows  A,  B,  C,  D,  •  •  •  and  A', 
B',  C,  D',  •  •  •  be  superposed  on  the  line  u.  Project 
them  both  to  a  point  8  and  pass  any  conic  k  through  S. 
We  thus  obtain  two  projective  pencils,  a,  b,  c,  d,  •  •  •  and 


INVOLUTION  77 

a',  b',  c',  d',  •  •  •  at  S,  which  meet  the  conic  in  the  points 
a,  18,7,2,."  and  a',  £',  7',  S',  •  •  •  (Fig.  36).  Take  now 
7  as  the  center  of  a  pencil  projecting  the  points  a',  /3', 
8',  •  •  •,  and  take  7'  as  the  center  of  a  pencil  projecting 
the  points  a,  f3,  8,  .  .  ..  These  two  pencils  are  projective 
to  each  other,  and  since  they  have  a  self-corresponding 
ray  in  common,  they  are  in  perspective  position  and 
corresponding  rays  meet  on  the  line  joining  (7a',  7'a) 
to  (7/3',  7'/3).  The  correspondence  between  points  in 
the  two  point-rows  on  u  is  now  easily  traced. 

130.  Application  of  Steiner's  construction  to  double 
correspondence.  Steiner's  construction  throws  into  our 
hands  an  important  theorem  concerning  double  corre- 
spondence :  If  two  projective  point-rows,  superposed  on 
the  same  line,  have  one  pair  of  points  which  correspond 
to  each  other  doubly,  then  all  pairs  correspond  to  each 
other  doubly,  and  the  line  is  paired  in  involution.  To 
make  this  appear,  let  us  call  the  point  A  on  u  by  two 
names,  A  and  P',  according  as  it  is  thought  of  as 
belonging  to  the  one  or  to  the  other  of  the  two  point- 
rows.  If  this  point  is  one  of  a  pair  which  correspond  to 
each  other  doubly,  then  the  points  A'  and  P  must  coin- 
cide (Fig.  37).  Take  now  any  point  C,  which  we  will 
also  call  B'.  We  must  show  that  the  corresponding 
point  C  must  also  coincide  with  the  point  E.  Join  all 
the  points  to  S,  as  before,  and  it  appears  that  the  points 
a  and  ir'  coincide,  as  also  do  the  points  a'ir  and  yp'. 
By  the  above  construction  the  line  y'p  must  meet  yp' 
on  the  line  joining  Qya',  7'a)  with  (77r',  7'7r).  But  these 
four  points  form  a  quadrangle  inscribed  in  the  conic, 
and  we  know  by  §  95  that  the  tangents  at  the  opposite 


78  PROJECTIVE  GEOMETRY 

vertices  7  and  7'  meet  on  the  line  v.  The  line  y'p 
is  thus  a  tangent  to  the  conic,  and  C  and  R  are 
the  same  point.  That  two  projective  point-rows  super- 
posed on  the  same  line  are  also  in  involution  when 
one  pair,  and  therefore  all  pairs,  correspond  doubly 
may  be  shown  by  taking  8  at  one  vertex  of  a  complete 


Fig.  37 

quadrangle  which  has  two  pairs  of  opposite1  sides  going 
through  two  pairs  of  points.  The  details  we  leave  to 
the  student. 

131.  Involution  of  points  on  a  point-row  of  the  second 
order.  It  is  important  to  note  also,  in  Steiner's  con- 
struction, that  we  have  obtained  two  point-rows  of  the 
second  order  superposed  on  the  same  conic,  and  have 
paired  the  points  of  one  with  the  points  of  the  other 
in  such  a  way  that  the  correspondence  is  double.  We 
may  then  extend  the  notion  of  involution  to  point-rows 
of  the  second  order  and  say  that  the  points  of  a  conic 
are  paired   in   involution   when   they   are    corresponding 


INVOLUTION  79 

points  of  two  projective  point-rows  superposed  on  the  conic, 
and  when  they  correspond  to  each  other  doubly.  With  this 
definition  we  may  prove  the  theorem  :  The  lines  joining 
corresponding  points  of  a  point-row  of  tlie  second  order  in 
involution  all  pass  through  a  fixed  point  U,  and  the  line 
joining  any  two  points  A,  B  meets  the  line  joining  the 
two  corresponding  points  A',  B'  in  the 
points  of  a  line  u,  which  is  the  polar 
of  U  with  respect  to  the  conic.  For 
take  A  and  A'  as  the  centers  of  two 
pencils,  the  first  perspective  to  the 
point-row  A',  B',  C'  and  the  second 
perspective  to  the  point-row  A,  B,  C. 
Then,  since  the  common  ray  of  the 
two  pencils  corresponds  to  itself,  they  are  in  perspec- 
tive position,  and  their  axis  of  perspectivity  u  (Fig.  38) 
is  the  line  which  joins  the  point  (AB1,  A'B~)  to  the 
point  (AC',  A'C~).  It  is  then  immediately  clear,  from 
the  theory  of  poles  and  polars,  that  BB'  and  CC'  pass 
through  the  pole  U  of  the  line  u. 

132.  Involution  of  rays.  The  whole  theory  thus  far 
developed  may  be  dualized,  and  a  theory  of  lines  in 
involution  may  be  built  up,  starting  with  the  complete 
quadrilateral.    Thus, 

The  three  pairs  of  rays  which  may  be  drawn  from  a 
point  through  the  three  pairs  of  opposite  vertices  of  a 
complete  quadrilateral  are  said  to  be  in  involution.  If  the 
pairs  aa'  and  bb'  are  fixed,  and  the  line  c  describes  a  pencil, 
the  corresponding  line  c'  also  describes  a  pencil,  and  the 
rays  of  the  pencil  are  said  to  be  paired  in  the  involution 
determined  by  aa'  and  bb'. 


80  PROJECTIVE  GEOMETRY 

133.  Double  rays.  The  self-corresponding  rays,  of 
which  there  are  two  or  none,  are  called  double  rays  of 
the  involution.  Corresponding  rays  of  the  involution 
are  harmonic  conjugates  with  respect  to  the  double 
rays.  To  the  theorem  of  Desargues  (§  125)  which  has 
to  do  with  the  system  of  conies  through  four  points 
we  have  the  dual : 

The  tangents  from  a  fixed  point  to  a  system  of  conies  t<m- 
gent  to  four  fixed  lines  form  a  pencil  of  rays  in  involution. 

134.  If  a  conic  of  the  system  should  go  through  the 
fixed  point,  it  is  clear  that  the  two  tangents  would  co- 
incide and  indicate  a  double  ray  of  the  involution.  The 
theorem,  therefore,  follows: 

Two  conies  or  none  may  be  drawn  through  a  fixed  point 
to  be  tangent  to  four  fixed  lines. 

135.  Double  correspondence.  It  further  appears  that 
two  projective  pencils  of  rays  which  have  the  same 
center  are  in  involution  if  one  pair  of  rays  correspond 
to  each  other  doubly.  From  this  it  is  clear  that  \\v 
might  have  defined  six  rays  in  involution  as  six  rays 
which  pass  through  a  point  and  also  through  six  points 
in  involution.  While  this  would  have  been  entirely  in 
accord  with  the  treatment  which  was  given  the  corre- 
sponding problem  in  the  theory  of  harmonic  points  and 
lines,  it  is  more  satisfactory,  from  an  aesthetic  point  of 
view,  to  build  the  theory  of  lines  in  involution  on  its  own 
base.  The  student  can  show,  by  methods  entirely  analo- 
gous to  those  used  in  the  second  chapter,  that  involution 
is  a  projective  property;  that  is.  six  rays  in  involution  are 
cut  by  any  transversal  in  six  points  in  involution. 


INVOLUTION  81 

136.  Pencils  of  rays  of  the  second  order  in  involution. 
We  may  also  extend  the  notion  of  involution  to  pen- 
cils of  rays  of  the  second  order.  Thus,  the  tangents  to  a 
conic  are  in  involution  when  they  are  corresponding  rays 
of  two  projective  pencils  of  the  second  order  superposed 
upon  the  same  conic,  and  when  they  correspond  to  each 
other  doubly.    We  have  then  the  theorem : 

137.  The  intersections  of  corresponding  rays  of  a  pen- 
cil of  the  second  order  in  involution  are  all  on  a  straight 
line  u,  and  the  intersection  of  any  two  tangents  ab,  when 
joined  to  the  intersection  of  the  corresponding  tangents  a'b', 
gives  a  line  which  passes  through  a  fixed  point  U,  the  pole 
of  the  line  u  with  respect  to  the  conic. 

138.  Involution  of  rays  determined  by  a  conic.    We 

have  seen  in  the  theory  of  poles  and  polars  (§  103) 
that  if  a  point  P  moves  along  a  line  m,  then  the  polar 
of  P  revolves  about  a  point.  This  pencil  cuts  out  on  m 
another  point-row  P',  projective  also  to  P.  Since  the 
polar  of  P  passes  through  P',  the  polar  of  P'  also  passes 
through  P,  so  that  the  correspondence  between  P  and 
P'  is  double.  The  two  point-rows  are  therefore  in  invo- 
lution, and  the  double  points,  if  any  exist,  are  the  points 
where  the  line  m  meets  the  conic.  A  similar  involution 
of  rays  may  be  found  at  any  point  in  the  plane,  corre- 
sponding rays  passing  each  through  the  pole  of  the  other. 
We  have  called  such  points  and  rays  conjugate  with 
respect  to  the  conic  (§  100).  We  may  then  state  the 
following  important  theorem : 

139.  A  conic  determines  on  every  line  in  its  plane  an 
involution  of  points,  corresponding  points  in  the  involution 


82  PROJECTIVE  GEOMETRY 

being  conjugate  with  respect  to  the  conic.    Tin-  double  points, 
if  any  exist,  are  the  points  where  the  lin.   meets  tlu 

140.  The  dual  theorem  reads:  A  c»iti<-  determines  ai 
every  point  in  the  plan*-  an  woohttion  of  rays,  correspond 
ing  rays  being  conjugate  with  respect  to  the  conic.  The 
double  rays,  if  any  exist,  are  ike  tangents  from  tin-  point 
to  the  conic. 

PROBLEMS 

1.  Two  lines  are  drawn  through  a  point  on  a  conic  so 
as  always  to  make  right  angles  with  each  other.  Show  that 
the  lines  joining  the  points  where  they  meet  the  conic  again 
all  pass  through  a  fixed  point. 

2.  Two  lines  are  drawn  through  a  fixed  point  on  a  conic 
so  as  always  to  make  equal  angles  with  the  tangent  at  that 
point.  Show  that  the  lines  joining  the  two  points  where  the 
lines  meet  the  conic  again  all  pass  through  a  fixed  point. 

3.  Four  lines  divide  the  plane  into  a  certain  number  of 
legions.  Determine  for  each  region  whether  two  conies  or 
none  may  be  drawn  to  pass  through  points  of  it  and  also 
to  be  tangent  to  the  four  lines.   (See  §  144.) 

4.  If  a  variable  quadrangle  move  in  such  a  way  as 
always  to  remain  inscribed  in  a  fixed  conic,  while  three  of 
its  sides  turn  each  around  one  of  three  fixed  oollinear  points, 
then  the  fourth  will  also  turn  around  a  fourth  fixed  point 
collinear  with  the  other  three. 

5.  State  and  prove  the  dual  of  problem  4. 

6.  Extend  problem  4  as  follows  :  If  a  variable  polygon  of 
an  even  number  of  sides  move  in  such  a  way  as  always  to 
remain  inscribed  in  a  fixed  conic,  while  all  its  sides  but  one 
pass  through  as  many  fixed  collinear  points,  then  the  last  side 
will  also  pass  through  a  fixed  point  collinear  with  the  others. 


INVOLUTION  83 

7.  If  a  triangle  QRS  be  inscribed  in  a  conic,  and  if  a 
transversal  s  meet  two  of  its  sides  in  A  and  A',  the  third 
side  and  the  tangent  at  the  opposite  vertex  in  B  and  B',  and 
the  conic  itself  in  C  and  C',  then  A  A',  BB',  CC  are  three 
pairs  of  points  in  an  involution. 

8.  Use  the  last  exercise  to  solve  the  problem  :  Given  five 
points,  Q,  R,  S,  C,  C',  on  a  conic,  to  draw  the  tangent  at  any 
one  of  them. 

9.  State  and  prove  the  dual  of  problem  7  and  use  it  to 
prove  the  dual  of  problem  8. 

10.  If  a  transversal  cut  two  tangents  to  a  conic  in  B  and 
B',  their  chord  of  contact  in  A,  and  the  conic  itself  in  P 
and  P',  then  the  point  A  is  a  double  point  of  the  involution 
determined  by  BB'  and  PP'. 

11.  State  and  prove  the  dual  of  problem  10. 

12.  If  a  variable  conic  pass  through  two  given  points, 
P  and  /'',  and  if  it  be  tangent  to  two  given  lines,  the  chord 
of  contact  of  these  two  tangents  will  always  pass  through 
one  of  two  fixed  points  on  PP'. 

13.  Use  the  last  theorem  to  solve  the  problem:  Given 
four  points,  P,  ]>',  Q,  S,  on  a  conic,  and  the  tangent  at  one  of 
them,  Q,  to  draw  the  tangent  at  any  one  of  the  other  points,  S. 

14.  Apply  the  theorem  of  problem  10  to  the  case  of  a 
hyperbola  where  the  two  tangents  are  the  asymptotes.  Show 
in  this  way  that  if  a  hyperbola  and  its  asymptotes  be  cut 
by  a  transversal,  the  segments  intercepted  by  the  curve  .and 
by  the  asymptotes  respectively  have  the  same  middle  point. 

15.  In  a  triangle  circumscribed  about  a  conic,  any  side  is 
divided  harmonically  by  its  point  of  contact  and  the  point 
where  it  meets  the  chord  joining  the  points  of  contact  of  the 
other  two  sides. 


CHAPTER  IX 

METRICAL  PROPERTIES  OF  INVOLUTIONS 

141.  Introduction  of  infinite  point;  center  of  involution. 
We  connect  the  projective  theory  of  involution  with  the 
metrical,  as  usual,  by  the  introduction  of  the  elements  at 
infinity.  In  an  involution  of  points  on  a  line  the  point 
which  corresponds  to  the  infinitely  distant  point  is  called 


Fig.  39 


the  center  of  the  involution.  Since  corresponding  points 
in  the  involution  have  been  shown  to  be  harmonic  con- 
jugates with  respect  to  the  double  points,  the  center  is 
midway  between  the  double  points  when  they  exist.  To 
construct  the  center  (Fig.  39)  we  draw  as  usual  through 
A  and  A'  any  two  rays  and  cut  them  by  a  line  parallel 
to  A  A'  in  the  points  K  and  31.  Join  these  points  to 
B  and  B\  thus  determining  on  AK  and  .4'iVthe  points  L 
and  N.   LN  meets  AA'  in  the  center  0  of  the  involution. 

84 


METRICAL  PROPERTIES  85 

142.  Fundamental  metrical  theorem.  From  the  figure 
we  see  that  the  triangles  OLB'  and  PLM  are  similar,  P 
being  the  intersection  of  KM  and  LN.  Also  the  tri- 
angles KPN  and  BON  are  similar.    We  thus  have 

OB:PK=ON:PN 
and  OB':PM=OL:PL; 

whence    OB  •  OB' :  PK  ■  PM=  ON  -OL:PN-  PL. 

In  the  same  way,  from  the  similar  triangles  OAL  and 
PKL,  and  also  OA' N  and  PMN,  we  obtain 

OA-  OA':PK-PM=ON-  OL-.PNPL, 
and  tli is,  with  the  preceding,  gives  at  once  the  funda- 
mental theorem,  which  is  sometimes  taken  also  as  the 
definition  of  involution : 

OA  •  OA'=OB  •  OB'  =  constant, 
or,  in  words, 

The  product  of  the  distances  from  the  center  to  two  cor- 
responding points  in  an  involution  of  points  is  constant. 

143.  Existence  of  double  points.  Clearly,  according 
as  the  constant  is  positive  or  negative  the  involution 
will  or  will  not  have  double  points.  The  constant  is 
the  square  of  the  distance  from  the  center  to  the 
double  points.  If  A  and  A'  lie  both  on  the  same  side 
of  the  center,  the  product  OA  •  OA'  is  positive ;  and  if 
they  lie  on  opposite  sides,  it  is  negative.  Take  the  case 
where  they  both  lie  on  the  same  side  of  the  center,  and 
take  also  the  pair  of  corresponding  points  BB'.  Then, 
since  OA  •  OA'  =  OB  ■  OB',  it  cannot  happen  that  B  and 
B'  are  separated  from  each  other  by  A  and  A'.  This  is 
evident  enough  if  the  points  are  on  opposite  sides  of 
the  center.    If  the  pairs  are  on  the  same  side  of  the 


86 


PROJECTIVE  GEOMETRY 


center,  and  B  lies  between  A  and  A',  so  that  OB  is 
greater,  say,  than  OA,  but  less  than  OAr,  then,  by  the 
equation  OA  ■  OA'  =  OB  •  OB',  we  must  have  OB'  also 
less  than  OA'  and  greater  than  OA.  A  similar  discus- 
sion may  be  made  for  the  ease  where  A  and  A'  lie  on 
opposite  sides  of  O.  The  results  may  be  stated  as 
follows,  without  any  reference  to  the  center: 

Given  two  pairs  of  points  in  <n>  involution  of  points,  if 
the  points  of  one  pair  are  separated  from  each  other  by 
the  points  of  the  other  pair,  then  the  involution  has  no 
double  points.  If  the  points  of  one  pair  are  not  separated 
from  each  other  by  the  points  of  the  other  pair,  then  the 
involution  has  two  double  points. 

144.  An  entirely  similar  criterion  decides  whether  an 
involution  of  rays  has  or  has  not  double  rays,  or  whether 
an  involution  of  planes  has  or  has  not  double  planes. 

145.  Construction  of  an  involution  by  means  of  circles. 
The  equation  just  derived,  OA  ■  OA'—OB  ■  <)I>',  indicates 
another  simple  way  in  which 
points  of  an  involution  of 
points  may  be  constructed. 
Through  A  and  A'  draw  any 
circle,  and  draw  also  any  cir- 
cle through  B  and  B'  to  cut 
the  first  in  the  two  points  G 
and  G'  (Fig.  40).  Then  any  circle  through  G  and  G' 
will  meet  the  line  in  pairs  of  points  in  the  involution 
determined  by  AA'  and  BB '.  For  if  such  a  circle  meets 
the  line  in  the  points  CC",  then,  by  the  theorem  in  the 
geometry  of  the  circle  which  says  that  if  any  chord  is 


Fig.  m 


METRICAL  PROPERTIES  87 

drawn  through  a  fixed  point  within  a  circle,  the  product 
of  its  segments  is  constant  in  whatever  direction  the  chord  is 
drawn,  and  if  a  secant  line  be  drawn  from  a  fixed  point 
without  a  circle,  the  product  of  the  secant  and  its  external 
segment  is  constant  in  whatever  direction  the  secant  line  is 
drawn,  we  have  OC  •  OC"=  OG  •  OG'  =  constant.  So  that 
for  all  such  points  OA  ■  OA'  =  <>B  •  OB'=OC  •  OC.  Fur- 
ther, the  line  GG'  meets  A  A'  in  the  center  of  the  invo- 
lution. To  find  the  double  points,  if  they  exist,  we  draw 
a  tangent  from  0  to  any  of  the  circles  through  GG'. 
Let  T  be  the  point  of  contact.  Then  lay  off  on  the 
line  OA  a  line  OF  equal  to  OT.  Then,  since  by  the  above 
theorem  of  elementary  geometry  OA  •  OA'  =  OT2  =  OF2, 
we  have  one  double  point  F.  The  other  is  at  an  equal 
distance  on  the  other  side  of  0.  This  simple  and  effec- 
tive method  of  constructing  an  involution  of  points  is 
often  taken  as  the  basis  for  the  theory  of  involution. 
In  projective  geometry,  however,  the  circle,  which  is  not 
a  figure  that  remains  unaltered  by  projection,  and  is 
essentially  a  metrical  notion,  ought  not  to  be  used  to 
build  up  the  purely  projective  part  of  the  theory. 

146.  It  ought  to  be  mentioned  that  the  theoiy  of 
analytic  geometry  indicates  that  the  circle  is  a  special 
conic  section  that  happens  to  pass  through  two  partic- 
ular imaginary  points  on  the  line  at  infinity,  called  the 
circular  points  and  usually  denoted  by  I  and  J.  The 
above  method  of  obtaining  a  point-row  in  involution  is, 
then,  nothing  but  a  special  case  of  the  general  theorem 
of  the  last  chapter  (§  125),  which  asserted  that  a  system 
of  conies  through  four  points  will  cut  any  line  in  the 
plane  in  a  point-row  in  involution. 


88  PROJECTIVE  GEOMETRY 

147.  Pairs  in  an  involution  of  rays  which  are  at  right 
angles.  Circular  involution.  In  an  involution  of  rays 
there  is  no  one  ray  which  may  be  distinguished  from 
all  the  others  as  the  point  at  infinity  is  distinguished 
from  all  other  points  on  a  line.  There  is  one  pair  of 
rays,  however,  which  does  differ  from  all  the  others  in 
that  for  this  particular  pair  the  angle  is  a  right  angle. 
This  is  most  easily  shown  by  using  the  construction 
that  employs  circles,  as  indicated  above.  The  centers  of 
all  the  circles  through  G  and  G'  lie  on  the  perpendicular 
bisector  of  the  line  GG'.  Let 
this  line  meet  the  line  AA' 
in  the  point  C  (Fig.  41),  and 
draw  the  circle  with  center  C 
which  goes  through  G  and  G'.  v  TT 
This  circle  cuts  out  two  points 
M  and  M '  in  the  involution.  The  rays  GM  and  GM'  are 
clearly  at  right  angles,  being  inscribed  in  a  semicircle. 
If,  therefore,  the  involution  of  points  is  projected  to 
G,  we  have  found  two  corresponding  rays  which  are 
at  right  angles  to  each  other.  Given  now  any  invo- 
lution of  rays  with  center  G,  we  may  cut  across  it 
by  a  straight  line  and  proceed  to  find  the  two  points 
M  and  M' .  Clearly  there  will  be  only  one  such  pair 
unless  the  perpendicular  bisector  of  GG'  coincides  with 
the  line  AA'.  In  this  case  every  ray  is  at  right  angles 
to  its  corresponding  ray,  and  the  involution  is  called 
circular. 

148.  Axes  of  conies.  At  the  close  of  the  last  chapter 
(§  140)  we  gave  the  theorem :  A  conic  determines  at  every 
point  in  ite  plane  an  involution  of  ruys,  oorrespondiitg  rays 


METRICAL  PROPERTIES  89 

being  conjugate  with  respect  to  the  conic.  The  double  rags, 
if  ang  exist,  are  the  tangents  from  the  point  to  the  conic. 
In  particular,  taking  the  point  as  the  center  of  the 
conic,  we  find  that  conjugate  diameters  form  a  system 
of  rays  in  involution,  of  which  the  asymptotes,  if  there 
are  any,  are  the  double  rays.  Also,  conjugate  diameters 
are  harmonic  conjugates  with  respect  to  the  asymptotes. 
By  the  theorem  of  the  last  paragraph,  there  are  two 
conjugate  diameters  which  are  at  right  angles  to  each 
other.  These  are  called  axes.  In  the  case  of  the  parab- 
ola, where  the  center  is  at  infinity,  and  on  the  curve, 
there  are,  properly  speaking,  no  conjugate  diameters. 
While  the  line  at  infinity  might  be  considered  as  con- 
jugate to  all  the  other  diameters,  it  is  not  possible  to 
assign  to  it  any  particular  direction,  and  so  it  cannot  be 
used  for  the  purpose  of  defining  an  axis  of  a  parabola. 
There  is  one  diameter,  however,  which  is  at  right  angles 
to  its  conjugate  system  of  chords,  and  this  one  is  called 
the  axis  of  the  parabola.  The  circle  also  furnishes  an 
exception  in  that  every  diameter  is  an  axis.  The  invo- 
lution in  this  case  is  circular,  every  ray  being  at  right 
angles  to  its  conjugate  ray  at  the  center. 

149.  Points  at  which  the  involution  determined  by 
a  conic  is  circular.  It  is  an  important  problem  to  dis- 
cover whether  for  any  conic  other  than  the  circle  it  is 
possible  to  find  any  point  in  the  plane  where  the  invo- 
lution determined  as  above  by  the  conic  is  circular. 
We  shall  proceed  to  the  curious  problem  of  proving  the 
existence  of  such  points  and  of  determining  their  num- 
ber and  situation.  We  shall  then  develop  the  important 
properties  of  such  points. 


90  PROJECTIVE  GEOMETRY 

150.  It  is  clear,  in  the  first  place,  that  such  a  point 
cannot  be  on  the  outside  of  the  conic,  else  the  involu- 
tion would  have  double  rays  and  such  rays  would  have 
to  be  at  right  angles  to  themselves.  In  the  second 
place,  if  two  such  points  exist,  the  line  joining  them 
must  be  a  diameter  and,  indeed,  an  axis.  For  if  F 
and  F'  were  two  such  points,  then,  since  the  conjugate 
ray  at  F  to  the  line  FF'  must  be  at  right  angles  to  it, 
and  also  since  the  conjugate  ray  at  F'  to  the  line  FF' 
must  be  at  right  angles  to  it,  the  pole  of  FF'  must 
be  at  infinity  in  a  direction  at  right  angles  to  FF'. 
The  line  FF'  is  then  a  diameter,  and  since  it  is  at 
right  angles  to  its  conjugate  diameter,  it  must  be  an 
axis.  From  this  it  follows  also  that  the  points  we  are 
seeking  must  all  lie  on  one  of  the  two  axes,  else  we 
should  have  a  diameter  which  does  not  go  through 
the  intersection  of  the  axes  —  the  center  of  the  conic. 
At  least  one  axis,  therefore,  must  be  free  from  any 
such  points. 

151.  Let  now  P  be  a  point  on  one  of  the  axes  (Fig.  42), 
and  draw  any  ray  through  it,  such  as  q.  As  q  revolves 
about  P,  its  pole  Q  moves  along  a  line  at  right  angles 
to  the  axis  on  which  P  lies,  describing  a  point-row  p 
projective  to  the  pencil  of  rays  q.  The  point  at  infinity 
in  a  direction  at  right  angles  to  q  also  describes  a  point- 
row  projective  to  q.  The  line  joining  corresponding- 
points  of  these  two  point-rows  is  always  a  conjugate 
line  to  q  and  at  right  angles  to  q,  or,  as  we  may  call  it, 
a  conjugate  normal  to  q.  These  conjugate  normals  to  q, 
joining  as  they  do  corresponding  points  in  two  projec- 
tive point-rows,  form   a  pencil  of  rays  of  the  second 


METRICAL  PROPERTIES 


91 


order.  But  since  the  point  at  infinity  on  the  point-row 
Q  corresponds  to  the  point  at  infinity  in  a  direction 
at  right  angles  to  q,  these  point-rows  are  in  perspec- 
tive position  and  the  normal  conjugates  of  all  the  lines 
through  P  meet  in  a  point.  This  point  lies  on  the 
same  axis  with  P,  as  is  seen  by  taking  q  at  right  angles 
to  the  axis  on  which  P  lies.  The  center  of  this  pencil 
may  be  called  P',  and  thus  we  have  paired  the  point  P 
with  the  point  P'.  By  moving  the  point  P  along  the 
axis,  and  by  keeping  the 
ray  q  parallel  to  a  fixed 
direction,  we  may  see  that 
the  point-row  P  and  the 
point-row  P'  are  projective. 
Also  the  correspondence  is 
double,  and  by  starting 
from  the  point  P'  we  arrive 
at  the  point  P.  Therefore 
the  point-rows  P  and  P'  are 
in   involution,    and  if  only 

the  involution  has  double  points,  we  shall  have  found 
in  them  the  points  we  are  seeking.  For  it  is  clear  that 
the  rays  through  P  and  the  corresponding  rays  through 
P'  are  conjugate  normals ;  and  if  P  and  P'  coincide,  we 
shall  have  a  point  where  all  rays  are  at  right  angles 
to  their  conjugates.  We  shall  now  show  that  the  invo- 
lution thus  obtained  on  one  of  the  two  axes  must  have 
double  points. 

152.  Discovery  of  the  foci  of  the  conic.  We  know 
that  on  one  axis  no  such  points  as  we  are  seeking  can 
lie  (§  150).    The  involution  of  points  PP'  on  this  axis 


Fie.  42 


92  PROJECTIVE  GEOMETRY 

can  therefore  have  no  double  points.  Nevertheless,  let 
PP'  and  RE'  be  two  pairs  of  corresponding  points  on 
this  axis  (Fig.  43).  Then  we  know  that  P  and  P'  are 
separated  from  each  other  by  R  and  R'  (§  143).  Draw 
a  circle  on  PP'  as  a  diameter,  and  one  on  RI?'  as  a 
diameter.  These  must  intersect  in 
two  points,  F  and  F',  and  since  the 
center  of  the  conic  is  the  center 
of  the  involution  PP',  RR',  as  is 
easily  seen,  it  follows  that  F  and  F' 
are  on  the  other  axis  of  the  conic. 
Moreover,  FR  and  FR'  are  con- 
jugate normal  rays,  since  RFR'  is 
inscribed  in  a  semicircle,  and  the 
two  rays  go  one  through  R  and  the  other  through  R'. 
The  involution  of  points  PP',  RR'  therefore  projects 
to  the  two  points  F  and  F'  in  two  pencils  of  rays  in 
involution  which  have  for  corresponding  rays  conjugate 
normals  to  the  conic.    We  may,  then,  say: 

There  are  two  and  only  two  points  of  the  plane  where 
the  involution  determined  by  the  conic  is  circular.  Tins,' 
two  points  lie  on  one  of  the  axes,  at  equal  distances  from 
the  center,  on  the  inside  of  the  conic.  These  points  are 
called  the  foci  of  the  came 

153.  The  circle  and  the  parabola.  The  above  dis- 
cussion applies  only  to  the  central  conies,  apart  from 
the  circle.  In  the  circle  the  two  foci  fall  together  at  the 
center.  In  the  case  of  the  parabola,  that  part  of  the 
investigation  which  proves  the  existence  of  two  foci  on 
one   of   the   axes  will   not  hold,  as  we   have   but   one 


METRICAL  PROPERTIES  93 

axis.  It  is  seen,  however,  that  as  /'  moves  to  infinity, 
carrying  the  line  q  with  it,  q  becomes  the  line  at  infin- 
ity, which  for  the  parabola  is  a  tangent  line.  Its  pole 
Q  is  thus  at  infinity  and  also  the  point  P',  so  that  P 
and  P'  fall  together  at  infinity,  and  therefore  one  focus 
of  the  parabola  is  at  infinity.  There  must  therefore  be 
another,  so  that 

A  parabola  hast  one  and  only  one  foeus  in  the  finite 
part  of  the  plane. 

154.  Focal  properties  of  conies.  We  proceed  to  de- 
velop some  theorems  which  will  exhibit  the  importance 
of  these  points  in  the  theory  of  the  conic  section. 
Draw  a  tangent  to  the  conic,  and  also  the  normal 
at  the  point  of  contact  P.  These 
two  lines  are  clearly  conjugate 
normals.  The  two  points  T  and 
X,  therefore,  where  they  meet  the 
axis  which  contains  the  foci,  are 
corresponding  points  in  the  invo- 
lution  considered   above,   and  are 

therefore  harmonic  conjugates  with  respect  to  the  foci 
(Fig.  44);  and  if  we  join  them  to  the  point  P,  we 
shall  obtain  four  harmonic  lines.  But  two  of  them 
are  at  right  angles  to  each  other,  and  so  the  others 
make  equal  angles  with  them  (Problem  4,  Chapter  II). 
Therefore 

The  linen  joining  a  point  on  the  conic  to  the  foci  make 
equal  angles  with  the  tangent. 

It  follows  that  rays  from  a  source   of  light  at  one 
foeus  are  reflected  by  an  ellipse  to  the  other. 


94  PROJECTIVE  GEOMETRY 

155.  In  the  case  of  the  parabola,  where  one  of  the 
foci  must  be  considered  to  be  at  infinity  in  the  direction 
of  the  diameter,  we  have 

A  diameter  makes  the  same 
angle  with  the  tangent  at  its 
extremity  as  that  tangent  does 
with  the  line  from  its  point  of 
contact  to  the  focus  (Fig.  4.")).  Fig.  45 

156.  This  last  theorem  is  the  basis  for  the  construc- 
tion of  the  parabolic  reflector.  A  ray  of  light  from  the 
focus  is  reflected  from  such  a  reflector  in  a  direction 
parallel  to  the  axis  of  the  reflector. 

157.  Directrix.  Principal  axis.  Vertex.  The  polar  of 
the  focus  with  respect  to  the  conic  is  called  the  directrix* 
The  axis  which  contains  the  foci  is  called  the  principal 
axis,  and  the  intersection  of  the  axis  with  the  curve  is 
called  the  vertex  of  the  curve.  The  directrix  is  at  right 
angles  to  the  principal  axis.  In  a  parabola  the  vertex 
is  equally  distant  from  the  focus  and  the  directrix, 
these  three  points  and  the  point  at  infinity  on  the  axis 
being  four  harmonic  points.  In  the  ellipse  the  vertex  is 
nearer  to  the  focus  than  it  is  to  the  directrix,  for  the 
same  reason,  and  in  the  hyperbola  it  is  farther  from 
the  focus  than  it  is  from  the  directrix. 

158.  Another  definition  of  a  conic.  Let  P  be  any  point 
on  the  directrix  through  which  a  line  is  drawn  meeting 
the  conic  in  the  points  A  and  B  (Fig.  46).  Let  the  tan- 
gents at  A  and  B  meet  in  1\  and  call  the  focus  F.  Then 
TF  and  PF  are  conjugate  lines,  and  as  they  pass  through 
a  focus  they  must  be  at  right  angles  to  each  other.     Let 


METRICAL  PROPERTIES 


95 


p 

1/ 

A' 
M 

/    F  / 

B' 

N 

■^L^ 

■^~^^B^~~r~~~^ 

Fig.  40 

TF  meet  AB  in  C.  Then  P,  A,  C,  B  are  four  harmonic 
points.  Project  these  four  points  parallel  to  TF  upon 
the  directrix,  and  we  then  get 
the  four  harmonic  points  P, 
M,  Q,  iV".  Since,  now,  TFP  is 
a  right  angle,  the  angles  MFQ 
and  NFQ  are  equal,  as  well 
as  the  angles  AFC  and  BFC. 
Therefore  the  triangles  MAF 
and  XFB  are  similar,  and 
FA :  AM=  FB :  BN.  Dropping 
perpendiculars  A  A'  and  BB' 
upon  the  directrix,  this  be- 
comes FA :  A  A'  =  FB :  BB1.  We 

have   thus  the   property   often  taken  as   the   definition 
of  a  conic : 

The  ratio  of  the  distances  from  a  point  on  the  conic  to 
the  fociis  and  the  directrix  is  constant. 

159.  Eccentricity.  By  taking  the  point  at  the  vertex 
of  the  conic,  we  note  that  this  ratio  is  less  than  unity 
for  the  ellipse,  greater  than  unity  for  the  hyperbola, 
and  equal  to  unity  for  the  pa- 
rabola. This  ratio  is  called  the 
eccentricity. 

160.  Sum  or  difference  of  focal 
distances.  The  ellipse  and  the 
hyperbola  have  two  foci  and 
two  directrices.  The  eccentricity,  of  course,  is  the  same 
for  one  focus  as  for  the  other,  since  the  curve  is  sym- 
metrical with  respect  to  both.    If  the   distances  from 


96 


PROJECTIVE  GEOMETRY 


Fig.  48 


a  point  on  a  conic  to  the  two  foci  are  r  and  r',  and 
the  distances  from  the  same  point  to  the  corresponding 
directrices  are  d  and  d' 
(Fig.  47),  we  have  r:d  = 
r''.d'=(r±rJy.(d±d'').  In  the 
ellipse  (d  +  d'}  is  constant, 
being  the  distance  between 
the  directrices.  In  the  hyper- 
bola this  distance  is  (d  —  ef  ). 
It  follows  (Fig.  48)  that 

In  the  ellipse  the  sum  of  the 
focal  distances  of  any  point 
on  the  curve  is  constant,  and 
in  the  hyperbola  the  difference  between  the  focal  distances 
is  constant. 

PROBLEMS 

1.  Construct  the  axis  of  a  parabola,  given  four  tangents. 

2.  Given  two  conjugate  lines  at  right  angles  to  each 
other,  and  let  them  meet  the  axis  which  has  no  foci  on  it 
in  the  points  A  and  B.  The  circle  on  AB  as  diameter  will 
pass  through  the  foci  of  the  conic. 

3.  Given  the  axes  of  a  conic  in  position,  and  also  a 
tangent  with  its  point  of  contact,  to  construct  the  foci  and 
determine  the  length  of  the  axes. 

4.  Given  the  tangent  at  the  vertex  of  a  parabola,  and 
two  other  tangents,  to  find  the  focus. 

5.  The  locus  of  the  center  of  a  circle  touching  two  given 
circles  is  a  conic  with  the  centers  of  the  given  circles  for 
its  foci. 

6.  Given  the  axis  of  a  parabola  and  a  tangent,  with  its 
point  of  contact,  to  find  the  focus. 


METRICAL  PROPERTIES  97 

7.  The  locus  of  the  center  of  a  circle  which  touches  a 
given  line  and  a  given  circle  consists  of  two  parabolas. 

8.  Let  F  and  F1  be  the  foci  of  an  ellipse,  and  P  any 
point  on  it.  Produce  FP  to  G,  making  PG  equal  to  PF'. 
Find  the  locus  of  G. 

9.  If  the  points  G  of  a  circle  be  folded  over  upon  a 
point  F,  the  creases  will  all  be  tangent  to  a  conic.  If  F  is 
within  the  circle,  the  conic  will  be  an  ellipse ;  if  F  is  without 
the  circle,  the  conic  will  be  a  hyperbola. 

10.  If  the  points  G  in  the  last  example  be  taken  on  a 
straight  line,  the  locus  is  a  parabola. 

11.  Find  the  foci  and  the  length  of  the  principal  axis  of 
the  conies  in  problems  9  and  10. 

12.  In  problem  10  a  correspondence  is  set  up  between 
straight  lines  and  parabolas.  As  there  is  a  fourfold  infinity 
of  parabolas  in  the  plane,  and  only  a  twofold  infinity  of 
straight  lines,  there  must  be  some  restriction  on  the  par- 
abolas obtained  by  this  method.  Find  and  explain  this 
restriction. 

13.  State  and  explain  the  similar  problem  for  problem  9. 

14.  The  last  four  problems  are  a  study  of  the  conse- 
quences of  the  following  transformation  :  A  point  O  is  fixed 
in  the  plane.  Then  to  any  point  P  is  made  to  correspond 
the  line  p  at  right  angles  to  OP  and  bisecting  it.  In  this 
correspondence,  what  happens  to  p  when  P  moves  along  a 
straight  line  ?  What  corresponds  to  the  theorem  that  two 
lines  have  only  one  point  in  common  ?  What  to  the  theorem 
that  the  angle  sum  of  a  triangle  is  two  right  angles  ?  Etc. 


CHAPTER  X 

ON  THE  HISTORY  OF  SYNTHETIC  PROJECTIVE  GEOMETRY 

161.  Ancient  results.  The  theory  of  synthetic  pro- 
jective geometry  as  we  have  built  it  up  in  this  course  is 
less  than  a  century  old.  This  is  not  to  say  that  many  of 
the  theorems  and  principles  involved  were  not  discov- 
ered much  earlier,  but  isolated  theorems  do  not  make  a 
theory,  any  more  than  a  pile  of  bricks  makes  a  building. 
The  materials  for  our  building  have  been  contributed 
by  many  different  workmen  from  the  days  of  Euclid 
down  to  the  present  time.  Thus,  the  notion  of  four 
harmonic  points  was  familiar  to  the  ancients,  who  con- 
sidered it  from  the  metrical  point  of  view  as  the  division 
of  a  line  internally  and  externally  in  the  same  ratio  *  ; 

*  The  more  general  notion  of  (inharmonic  ratio,  which  includes 
the  harmonic  ratio  as  a  special  case,  was  also  known  to  the  ancients. 
While  we  have  not  found  it  necessary  to  make  use  of  the  anharmoiuc 
ratio  in  building  up  our  theory,  it  is  so  frequently  met  with  in  treatises 
on  geometry  that  some  account  of  it  should  be  given. 

Consider  any  four  points,  A,  B,  C,  D,  on  a  line,  and  join  them  to 
any  point  S  not  on  that  line.  Then  the  triangles  ASB,  CSD,  ASD, 
CSB,  having  all  the  same  altitude,  are  to  each  other  as  their  bases. 
Also,  since  the  area  of  any  triangle  is  one  half  the  product  of  any  two 
of  its  sides  by  the  sine  of  the  angle  included  between  them,  we  have 

AB  x  CD_AS x  BS  sin  ASB  xCSxDS  sin CSD  _&in  ASBx  sinCSD 
AD  x  CB~AS  x  DS  sin  ASD  xCSxBS  sin  CSB~ sin  ASD  x  sin  CSB' 

Now  the  fraction  on  the  right  would  be  unchanged  if  instead  of  the 
points  A,  B,  C,  D  we  should  take  any  other  four  points  A',  B',  C,  D" 
lying  on  any  other  line  cutting  across  SA,  SB,  SC,  SD.    In  other 

98 


SYNTHETIC  PKOJECTIVE  GEOMETRY        99 

the  involution  of  six  points  cut  out  by  any  transversal 
which  intersects  the  sides  of  a  complete  quadrilateral 

words,  the  fraction  on  the  left  is  unaltered  in  value  if  the  points 
A,  B,  C,  D  are  replaced  by  any  other  four  points  perspective  to  them. 
Again,  the  fraction  on  the  left  is  unchanged  if  some  other  point  were 
taken  instead  of  <S.  In  other  words,  the  fraction  on  the  right  is 
unaltered  if  we  replace  the  four  lines  SA,  SB,  SC,  SD  by  any  other  four 
lines  perspective  to  them.  The  fraction  on  the  left  is  called  the  anhar- 
monic  ratio  of  the  four  points  A,  B,  C,  D ;  the  fraction  on  the  right 
is  called  the  anharmonic  ratio  of  the  four  lines  SA,  SB,  SC,  SD.  The 
anharmonic  ratio  of  four  points  is  sometimes  written  (ABCD),  so  that 

AB*CD  =  {ABCD). 
ADxCB      K  ' 

If  we  take  the  points  in  different  order,  the  value  of  the  anharmonic 
ratio  will  not  necessarily  remain  the  same.  The  twenty-four  different 
ways  of  writing  them  will,  however,  give  not  more  than  six  different 
values  for  the  anharmonic  ratio,  for  by  writing  out  the  fractions 
which  define  them  we  can  find  that  (ABCD)  =  (BADC)  =  (CDAB)  = 
(DCBA).  If  we  write  (ABCD)  =  a,  it  is  not  difficult  to  show  that 
the  six  values  are 

a  ;  1/a  ;  1  -  a  ;  1/(1  -a);  (a-  \)/a  ;  a/(a  -  1). 

The  proof  of  this  we  leave  to  the  student. 

It  A,  B,  C,  D  are  four  harmonic  points  (see  Fig.  6,  p.  22),  and  a  quad- 
rilateral KLMX  is  constructed  such  that  KL  and  MN  pass  through 
A,  KN  and  LM  through  C,  LN  through  B,  and  KM  through  D,  then, 
projecting  A,  B,  C,  D  from  L  upon  KM,  we  have  (ABCD)  =  (KOMD), 
where  O  is  the  intersection  of  KM  with  LN.  But,  projecting  again 
the  points  K,  0,  M,  D  from  N  back  upon  the  line  AB,  we  have 
(KOMD)  =  (CBAD).   From  this  we  have 

(ABCD)  =  (CBAD), 

or  a  =  1/a ; 

whence  a  =  l  or  a  =  —  1.  But  it  is  easy  to  see  that  a  =  1  implies  that 
two  of  the  four  points  coincide.  For  four  harmonic  points,  therefore, 
the  six  values  of  the  anharmonic  ratio  reduce  to  three,  namely,  2,  -J-, 
and  —1.  Incidentally  we  see  that  if  an  interchange  of  any  two 
points  in  an  anharmonic  ratio  does  not  change  its  value,  then  the 
four  points  are  harmonic. 

Many  theorems  of  projective  geometry  are  succinctly  stated  in 
terms  of  anharmonic  ratios.    Thus,  the  anharmonic  ratio  of  any  four 


100  PROJECTIVE  GEOMETRY 

was  studied  by  Pappus*  ;  but  these  notions  were  not 
made  the  foundation  for  any  general  theory.  Taken  by 
themselves,  they  are  of  small  consequence ;  it  is  their 
relation  to  other  theorems  and  sets  of  theorems  that 
gives  them  their  importance.  The  ancients  were  doubt- 
less familiar  with  the  theorem,  Two  lines  determine  a 
j>"ii(t,  and  two  points  determine  a  line,  but  they  had 
no  glimpse  of  the  wonderful  law  of  duality,  of  which 
this  theorem  is  a  simple  example.  The  principle  of 
projection,  by  which  many  properties  of  the  conic  sec- 
tions may  be  inferred  from  corresponding  properties 
of  the  circle  which  forms  the  base  of  the  cone  from 
which  they  are  cut  —  a  principle  so  natural  to  modern 
mathematicians  —  seems  not  to  have  occurred  to  the 
Greeks.    The  ellipse,  the  hyperbola,  and  the  parabola 

elements  of  a  form  is  equal  to  the  anharmonic  ratio  of  the  corresponding 

four  elements  in  any  form  protectively  related  to  it.    The  anharmonic 

ratio  of  the  lines  joining  any  four  fixed  points  on  a  conic  to  a  variable 

fifth  point  on  the  conic  is  constant.    Tlie 

locus  of  points  from  which  four  points 

in  a  plane  are  seen  along  four  rays  of 

constant  anharmonic  ratio  is  a  conic 

through  the  four  points.  We  leave  these 

theorems  for  the  student,  who  may 

also  justify  the  following  solution  of 

the  problem  :   Given  three  points  and 

a  certain  anharmonic  ratio,  to  find  a 

fourth  point  which  shall  have  with  the 

given  three  the  given  anliarmonic  ratio. 

Let  A ,  B,  D  be  the  three  given  points 

(Fig.  49).    On    any   convenient   line 

through  A  take  two  points  B'  and  Tf  Fig.  49 

such  that  AB'/AIf  is  equal  to  the 

given  anharmonic  ratio.    Join  BB'  and  DD'  and  let  the  two  lines 

meet  in  S.   Draw  through  S  a  parallel  to  AR.   This  line  will  meet 

AB  in  the  required  point  C. 

*  Pappus,  Mathematicae  Collectiones,  vii,  129. 


SYNTHETIC  PROJECTIVE  GEOMETRY      101 

were  to  them  entirely  different  curves,  to  be  treated 
separately  with  methods  appropriate  to  each.  Thus  the 
focus  of  the  ellipse  was  discovered  some  five  hundred 
years  before  the  focus  of  the  parabola!  It  was  not  till 
1522  that  Verner*  of  Xurnberg  undertook  to  demon- 
strate the  properties  of  the  conic  sections  by  means  of 
the  circle. 

162.  Unifying  principles.  In  the  early  years  of  the 
seventeenth  century  —  that  wonderful  epoch  in  the 
history  of  the  world  which  produced  a  Galileo,  a  Kep- 
ler, a  Tycho  Brahe,  a  Descartes,  a  Desargues,  a  Pascal, 
a  Cavalieri,  a  Yv'allis,  a  Fermat,  a  Huygens,  a  Bacon, 
a  Napier,  and  a  goodly  array  of  lesser  lights,  to  say 
nothing  of  a  Rembrandt  or  of  a  Shakespeare  —  there 
began  to  appear  certain  unifying  principles  connecting 
the  great  mass  of  material  dug  out  by  the  ancients. 
Thus,  in  160-4  the  great  astronomer  Kepler  f  introduced 
the  notion  that  parallel  lines  should  be  considered  as 
meeting  at  an  infinite  distance,  and  that  a  parabola  is  at 
once  the  limiting  case  of  an  ellipse  and  of  a  hyperbola. 
He  also  attributes  to  the  parabola  a  "  blind  focus " 
(caecus  focus)  at  infinity  on  the  axis. 

163.  Desargues.  In  1639  Desargues,^  an  architect  of 
Lyons,  published  a  little  treatise  on  the  conic  sections, 
in  which  appears  the  theorem  upon  which  we  have 
founded   the   theory   of   four  harmonic    points   (§  25). 

*  J.  Verneri,  Libellus  super  vigintiduobus  elementisconicis,  etc.  1522. 

t  Kepler,  Ad  Vitellionem  paralipomena  quibus  astronomiae  pars 
optica  traditur.    1604. 

X  Desargues,  Bruillon-project  d'une  atteiute  aux  e^nements  des 
rencontres  d'un  cdne  avec  un  plan.  1639.  Edited  and  analyzed  by 
Poudra,  1864. 


102  PROJECTIVE  GEOMETRY 

Desargues,  however,  does  not  make  use  of  it  for  that 
purpose.  Four  harmonic  points  are  for  him  a  special 
case  of  six  points  in  involution  when  two  of  the  three 
pairs  coincide  giving  double  points.  His  development 
of  the  theory  of  involution  is  also  different  from  the 
purely  geometric  one  which  we  have  adopted,  and  is 
based  on  the  theorem  (§  142)  that  the  product  of  the 
distances  of  two  conjugate  points  from  the  center  is 
constant.  He  also  proves  the  projective  character  of 
an  involution  of  pomts  by  showing  that  when  six  lines 
pass  through  a  point  and  through  six  points  in  involu- 
tion, then  any  transversal  must  meet  them  in  six  points 
which  are  also  in  involution. 

164.  Poles  and  polars.  In  this  little  treatise  is  also 
contained  the  theory  of  poles  and  polars.  The  polar 
line  is  called  a  traversal.*  The  harmonic  properties  of 
poles  and  polars  are  given,  but  Desargues  seems  not 
to  have  arrived  at  the  metrical  properties  which  result 
when  the  infinite  elements  of  the  plane  are  introduced. 
Thus  he  says,  "  When  the  traversal  is  at  an  infinite 
distance,  all  is  unimaginable." 

165.  Desargues's  theorem  concerning  conies  through 
four  points.  We  find  in  this  little  book  the  beautiful 
theorem  concerning  a  quadrilateral  inscribed  in  a  conic 
section,  which  is  given  by  his  name  in  §  138.  The 
theorem  is  not  given  in  terms  of  a  system  of  conies 
through  four  points,  for  Desargues  had  no  conception  of 

*  The  term  'pole'  was  first  introduced,  in  the  sense  in  which  we 
have  used  it,  in  1810,  by  a  French  mathematician  named  Servois 
(Gergonne,  Annates  des  MatMmatiques,  I,  337),  and  the  corresponding 
term  '  polar '  by  the  editor,  Gergonne,  of  this  same  journal  three  years 
later. 


SYNTHETIC  PROJECTIVE  GEOMETRY      103 

any  such  system.  He  states  the  theorem,  in  effect,  as 
follows :  Given  a  simple  quadrilateral  inscribed,  in  a  conic 
section,  every  transversal  meets  the  conic  and  the  four  sides 
of  the  quadrilateral  in  six  points  which  are  in  involution. 

166.  Extension  of  the  theory  of  poles  and  polars  to 
space.  As  an  illustration  of  his  remarkable  powers  of 
generalization,  we  may  note  that  Desargues  extended 
the  notion  of  poles  and  polars  to  space  of  three  dimen- 
sions for  the  sphere  and  for  certain  other  surfaces  of 
the  second  degree.  This  is  a  matter  which  has  not 
been  touched  on  in  this  book,  but  the  notion  is  not 
difficult  to  grasp.  If  we  draw  through  any  point  P  in 
space  a  line  to  cut  a  sphere  in  two  points,  A  and  B,  and 
then  construct  the  fourth  harmonic  of  P  with  respect  to 
A  and  B,  the  locus  of  this  fourth  harmonic,  for  various 
lines  through  P,  is  a  plane  called  the  polar  plane  of  P 
with  respect  to  the  sphere.  With  this  definition  and  theo- 
rem one  can  easily  find  dual  relations  between  points 
and  planes  in  space  analogous  to  those  between  points  and 
lines  in  a  plane.  Desargues  closes  his  discussion  of  this 
matter  with  the  remark,  "Similar  properties  may  be 
found  for  those  other  solids  which  are  related  to  the 
sphere  in  the  same  way  that  the  conic  section  is  to  the 
circle."  It  should  not  be  inferred  from  this  remark, 
however,  that  he  was  acquainted  with  all  the  different 
varieties  of  surfaces  of  the  second  order.  The  ancients 
were  well  acquainted  with  the  surfaces  obtained  by 
revolving  an  ellipse  or  a  parabola  about  an  axis.  Even 
the  hyperboloid  of  two  sheets,  obtained  by  revolving  the 
hyperbola  about  its  major  axis,  was  known  to  them, 
but  probably  not  the  hyperboloid  of  one  sheet,  which 


104  PROJECTIVE  GEOMETRY 

results  from  revolving  a  hyperbola  about  the  other 
axis.  Al\  the  other  solids  of  the  second  degree  were 
probably  unknown  until  their  discovery  by  Euler.* 

167.  Desargues  had  no  conception  of  the  conic  section 
as  the  locus  of  intersection  of  corresponding  rays  of  two 
projective  pencils  of  rays.  He  seems  to  have  tried  to 
describe  the  curve  by  means  of  a  pair  of  compasses, 
moving  one  leg  back  and  forth  along  a  straight  line 
instead  of  holding  it  fixed  as  in  drawing  a  circle.  He 
does  not  attempt  to  define  the  law  of  the  movement 
necessary  to  obtain  a  conic  by  this  means. 

168.  Reception  of  Desargues's  work.  Strange  to  say, 
Desargues's  immortal  work  was  heaped  with  the  most  vio- 
lent abuse  and  held  up  to  ridicule  and  scorn  !  "  Incredi- 
ble errors  !  Enormous  mistakes  and  falsities  !  Really  it 
is  impossible  for  anyone  who  is  familiar  with  the  science 
concerning  which  he  wishes  to  retail  his  thoughts,  to 
keep  from  laughing ! "  Such  were  the  comments  of  re- 
viewers and  critics.  Nor  were  his  detractors  altogether 
ignorant  and  uninstructed  men.  In  spite  of  the  devotion 
of  his  pupils  and  in  spite  of  the  admiration  and  friend- 
ship of  men  like  Descartes,  Fermat,  Mersenne,  and 
Roberval,  his  book  disappeared  so  completely  that  two 
centuries  after  the  date  of  its  publication,  when  the 
French  geometer  Chasles  wrote  his  history  of  geometry, 
there  was  no  means  of  estimating  the  value  of  the  work 
done  by  Desargues.  Six  years  later,  however,  in  1845, 
Chasles  found  a  manuscript  copy  of  the  "  Bruillon- 
project,"  made  by  Desargues's  pupil,  De  la  Hire. 

*  Euler,  Introductio  in  analysin  inlinitorum,  Appendix,  cap.  V. 
1748. 


SYNTHETIC  PROJECTIVE  GEOMETRY      105 

169.  Conservatism  in  Desargues's  time.  It  is  not  neces- 
sary to  suppose  that  this  eff  acement  of  Desargues's  work 
for  two  centuries  was  due  to  the  savage  attacks  of  his 
critics.  All  this  was  in  accordance  with  the  fashion  of 
the  time,  and  no  man  escaped  bitter  denunciation  who 
attempted  to  improve  on  the  methods  of  the  ancients. 
Those  were  days  when  men  refused  to  believe  that  a 
heavy  body  falls  at  the  same  rate  as  a  lighter  one,  even 
when  Galileo  made  them  see  it  with  their  own  eyes 
at  the  foot  of  the  tower  of  Pisa.  Could  they  not  turn 
to  the  exact  page  and  line  of  Aristotle  which  declared 
that  the  heavier  body  must  fall  the  faster !  "  I  have 
read  Aristotle's  writings  from  end  to  end,  many  times," 
wrote  a  Jesuit  provincial  to  the  mathematician  and 
astronomer,  Christoph  Scheiner,  at  Ingolstadt,  whose 
telescope  seemed  to  reveal  certain  mysterious  spots  on 
the  sun,  "  and  I  can  assure  you  I  have  nowhere  found 
anything  similar  to  what  you  describe.  Go,  my  son,  and 
tranquilize  yourself ;  be  assured  that  what  you  take  for 
spots  on  the  sun  are  the  faults  of  your  glasses,  or  of 
your  eyes."  The  dead  hand  of  Aristotle  barred  the 
advance  in  every  department  of  research.  Physicians 
would  have  nothing  to  do  with  Harvey's  discoveries 
about  the  circulation  of  the  blood.  "  Nature  is  accused 
of  tolerating  a  vacuum ! "  exclaimed  a  priest  when  Pas- 
cal began  his  experiments  on  the  Puy-de-Dome  to  show 
that  the  column  of  mercury  in  a  glass  tube  varied  in 
height  with  the  pressure  of  the  atmosphere. 

170.  Desargues's  style  of  writing.  Nevertheless,  author- 
ity counted  for  less  at  this  time  in  Paris  than  it  did  in 
Italy,  and  the  tragedy  enacted  in  Rome  when  Galileo 


106  PKOJECTIVE  GEOMETRY 

was  forced  to  deny  his  inmost  convictions  at  the  bid- 
ding of  a  brutal  Inquisition  could  not  have  been  staged 
in  France.  Moreover,  in  the  little  company  of  scientists 
of  which  Desargues  was  a  member  the  utmost  liberty 
of  thought  and  expression  was  maintained.  One  very 
good  reason  for  the  disappearance  of  the  work  of  De- 
sargues is  to  be  found  in  his  style  of  writing.  He  failed 
to  heed  the  very  good  advice  given  him  in  a  letter  from 
his  warm  admirer  Descartes.*  "  You  may  have  two  de- 
signs, both  very  good  and  very  laudable,  but  which  do 
not  require  the  same  method  of  procedure :  The  one  is 
to  write  for  the  learned,  and  show  them  some  new  prop- 
erties of  the  conic  sections  which  they  do  not  already 
know;  and  the  other  is  to  write  for  the  curious  un- 
learned, and  to  do  it  so  that  this  matter  which  until 
now  has  been  understood  by  only  a  very  few,  and  which 
is  nevertheless  very  useful  for  perspective,  for  paint- 
ing, architecture,  etc.,  shall  become  common  and  easy  to 
all  who  wish  to  study  them  in  your  book.  If  you  have 
the  first  idea,  then  it  seems  to  me  that  it  is  necessary 
to  avoid  using  new  terms ;  for  the  learned  are  already 
accustomed  to  using  those  of  Apollonius,  and  will  not 
readily  change  them  for  others,  though  better,  and  thus 
yours  will  serve  only  to  render  your  demonstrations 
more  difficult,  and  to  turn  away  your  readers  from  your 
book.  If  you  have  the  second  plan  in  mind,  it  is  cer- 
tain that  your  terms,  which  are  French,  and  conceived 
with  spirit  and  grace,  will  be  better  received  by  persons 
not  preoccupied  with  those  of  the  ancients.  .  .  .  But,  if 
you  have  that  intention,  you  should  make  of  it  a  great 
*  (Euvres  de  Desargues,  t.  II,  132. 


SYNTHETIC  PROJECTIVE  GEOMETRY      107 

volume :  explain  it  all  so  fully  and  so  distinctly  that 
those  gentlemen  who  cannot  study  without  yawning ; 
who  cannot  distress  their  imaginations  enough  to  grasp 
a  proposition  in  geometry,  nor  turn  the  leaves  of  a  book 
to  look  at  the  letters  in  a  figure,  shall  find  nothing  in 
your  discourse  more  difficult  to  understand  than  the 
description  of  an  enchanted  palace  in  a  fairy  story." 
The  point  of  these  remarks  is  apparent  when  we  note 
that  Desargues  introduced  some  seventy  new  terms  in 
his  little  book,  of  which  only  one,  involution,  has  sur- 
vived. Curiously  enough,  this  is  the  one  term  singled 
out  for  the  sharpest  criticism  and  ridicule  by  his  re- 
viewer, De  Beaugrand.*  That  Descartes  knew  the  char- 
acter of  Desargues's  audience  better  than  he  did  is  also 
evidenced  by  the  fact  that  De  Beaugrand  exhausted  his 
patience  in  reading  the  first  ten  pages  of  the  book. 

171.  Lack  of  appreciation  of  Desargues.  Desargues's 
methods,  entirely  different  from  the  analytic  methods 
just  then  being  developed  by  Descartes  and  Fermat, 
seem  to  have  been  little  understood.  "  Between  you 
and  me,"  wrote  Descartes  f  to  Mersenne,  "  I  can  hardly 
form  an  idea  of  what  he  may  have  written  concerning 
conies."  Desargues  seems  to  have  boasted  that  he  owed 
nothing  to  any  man,  and  that  all  his  results  had  come 
from  his  own  mind.  His  favorite  pupil,  De  la  Hire,  did 
not  realize  the  extraordinary  simplicity  and  generality 
of  his  work.  It  is  a  remarkable  fact  that  the  only  one 
of  all  his  associates  to  understand  and  appreciate  the 
methods  of  Desargues  should  be  a  lad  of  sixteen  years.' 

*  CEuvres  de  Desargues,  t.  II,  370. 
t  CEuvres  de  Descartes,  t.  II,  499. 


108  PROJECTIVE  GEOMETRY 

172.  Pascal  and  his  theorem.  One  does  not  have  to 
believe  all  the  marvelous  stories  of  Pascal's  admiring 
sisters  to  credit  him  with  wonderful  precocity.  We  have 
the  fact  that  in  1640,  when  he  was  sixteen  years  old, 
he  published  a  little  placard,  or  poster,  entitled  "Essay 
pour  les  conique,"  *  in  which  his  great  theorem  appears 
for  the  first  time.  His  manner  of  putting  it  may  be  a 
little  puzzling  to  one  who  has  only  seen  it  in  the  form 
given  in  this  book,  and  it  may  be  worth  while  for  the 
student  to  compare  the  two  methods  of  stating  it.  It  is 
given  as  follows :  "If  in  the  plane  of  M,  S,  Q  we  draw 
through  M  the  two  lines  MK  and  MV,  and  through  the 
point  S  the  two  lines  SK  and  SV,  and  let  K  be  the  inter- 
section of  MK  and  SK  ;  V  the  intersection  of  MV  and 
SV;  A  the  intersection  of  MA  and  SA  (A  is  the  inter- 
section of  SV  and  MK),  and  fi  the  intersection  of  MV 
and  SK ;  and  if  through  two  of  the  four  points  A,  K, 
p,  V,  which  are  not  in  the  same  straight  line  with  M  and 
S,  such  as  K  and  V,  we  pass  the  circumference  of  a  circle 
cutting  the  lines  MV,  MP,  SV,  SK  in  the  points  0,  P„ 
Q,  N ;  I  say  that  the  lines  MS,  NO,  PQ  are  of  the  same 
order.''''  (By  w  lines  of  the  same  order "  Pascal  means 
lines  which  meet  in  the  same  point  or  are  parallel.)  By 
projecting  the  figure  thus  described  upon  another  plane 
he  is  able  to  state  his  theorem  for  the  case  where  the 
circle  is  replaced  by  any  conic  section. 

173.  It  must  be  understood  that  the  "  Essay "  was 
only  a  resume  of  a  more  extended  treatise  on  conies 
which,  owing  partly  to  Pascal's  extreme  youth,  partly 
to  the  difficulty  of  publishing  scientific  works  in  those 

*  CEuvros  de  Pascal,  par  Brunschvig  et  Boutroux,  t.  I.  262, 


SYNTHETIC  PROJECTIVE  GEOMETRY      109 

days,  and  also  to  his  later  morbid  interest  in  religious 
matters,  was  never  published.  Leibniz  *  examined  a  copy 
of  the  complete  work,  and  has  reported  that  the  great 
theorem  on  the  mystic  hexagram  was  made  the  basis  of 
the  whole  theory,  and  that  Pascal  had  deduced  some  four 
hundred  corollaries  from  it.  This  would  indicate  that 
here  was  a  man  able  to  take  the  unconnected  materials 
of  projective  geometry  and  shape  them  into  some  such 
symmetrical  edifice  as  we  have  to-day.  Unfortunately 
for  science,  Pascal's  early  death  prevented  the  further 
development  of  the  subject  at  his  hands. 

174.  In  the  M  Essay "  Pascal  gives  full  credit  to 
Desargues,  saying  of  one  of  the  other  propositions, 
?t  We  prove  this  property  also,  the  original  discoverer  of 
which  is  M.  Desargues,  of  Lyons,  one  of  the  greatest 
minds  of  this  age  .  .  .  and  I  wish  to  acknowledge  that 
I  owe  to  him  the  little  which  I  have  discovered."  This 
acknowledgment  led  Descartes  to  believe  that  Pascal's 
theorem  should  also  be  credited  to  Desargues.  But  in 
the  scientific  club  which  the  young  Pascal  attended 
in  company  with  his  father,  who  was  also  a  scientist 
of  some  reputation,  the  theorem  went  by  the  name  of 
1  la  Pascalia,'  and  Descartes's  remarks  do  not  seem  to 
have  been  taken  seriously,  which  indeed  is  not  to  be 
wondered  at,  seeing  that  he  was  in  the  habit  of  giving 
scant  credit  to  the  work  of  other  scientific  investigators 
than  himself. 

175.  De  la  Hire  and  his  work.  De  la  Hire  added 
little  to  the  development  of  the  subject,  but  he  did  put 
into  print  much  of  what  Desargues  had  already  worked 

*  Chasles,  Histoire  de  la  G6om6trie,  70. 


110  PROJECTIVE  GEOMETRY 

out,  not  fully  realizing,  perhaps,  how  much  was  his 
own  and  how  much  he  owed  to  his  teacher.  Writing  in 
1679,  he  says,*  "I  have  just  read  for  the  first  time 
M.  Desargues's  little  treatise,  and  have  made  a  copy 
of  it  in  order  to  have  a  more  perfect  knowledge  of  it." 
It  was  this  copy  that  saved  the  work  of  his  master 
from  oblivion.  De  la  Hire  should  be  credited,  among 
other  things,  with  the  invention  of  a  method  by  which 
figures  in  the  plane  may  be  transformed  into  others 
of  the  same  order.  His  method  is  extremely  interest- 
ing, and  will  serve  as  an  exercise  for  the  student  in 
synthetic  projective  geometry.  It  is  as  follows:  Draw 
two  parallel  lines,  a  ami  l>,  and  select  a  point  P  in  their 
plane.  Through  any  point  M  of  the  plane  draw  a  line 
meeting  a  in  A  and  h  in  B.  Draw  a  line  through  B 
parallel  to  APy  and  let  it  meet  MI'  in  the  point  M*.  It 
may  be  shown  that  the  point  M'  thus  obtained  does  not 
depend  at  all  on  the  particular  rag  JfAB  used  in  deter* 
mining  it,  so  that  we  have  set  up  a  one-to-one  correspond  nee 
between  the  points  M  and  31'  in  the  plane.  The  student 
may  show  that  as  M  describes  a  point-row,  M'  describes 
a  point-row  projective  to  it  As  M  describes  a  conic, 
M'  describes  another  conic.  This  sort  of  correspon- 
dence is  called  a  collineati<>u.  It  will  be  found  that  the 
points  on  the  line  b  transform  into  themselves,  as  does 
also  the  single  point  P.  Points  on  the  line  a  trans- 
form into  points  on  the  line  at  infinity-  The  student 
should  remove  the  metrical  features  of  the  construction 
and  take,  instead  of  two  parallel  lines  a  and  b,  any 
two  lines  which  may  meet  in  a  finite  part  of  the  plane. 
*  CEuvres  de  Desargues,  t.  I,  231. 


SYNTHETIC  PROJECTIVE  GEOMETRY      111 

The  collineation  is  a  special  one  in  that  the  general 
one  has  an  invariant  triangle  instead  of  an  invariant 
point  and  line. 

176.  Descartes  and  his  influence.  The  history  of  syn- 
thetic projective  geometry  has  little  to  do  with  the  work 
of  the  great  philosopher  Descartes,  except  in  an  indirect 
way.  The  method  of  algebraic  analysis  invented  by 
him,  and  the  differential  and  integral  calculus  which 
developed  from  it,  attracted  all  the  interest  of  the 
mathematical  world  for  nearly  two  centuries  after 
Desargues,  and  synthetic  geometry  received  scant  atten- 
tion during  the  rest  of  the  seventeenth  century  and  for 
the  greater  part  of  the  eighteenth  century.  It  is  difficult 
for  moderns  to  conceive  of  the  richness  and  variety  of 
the  problems  which  confronted  the  first  workers  in  the 
calculus.  To  come  into  the  possession  of  a  method 
which  would  solve  almost  automatically  problems  which 
had  baffled  the  keenest  minds  of  antiquity ;  to  be  able 
to  derive  in  a  few  moments  results  which  an  Archimedes 
had  toiled  long  and  patiently  to  reach  or  a  Galileo  had 
determined  experimentally ;  such  was  the  happy  expe- 
rience of  mathematicians  for  a  century  and  a  half  after 
Descartes,  and  it  is  not  to  be  wondered  at  that  along 
with  this  enthusiastic  pursuit  of  new  theorems  in  anal- 
ysis should  come  a  species  of  contempt  for  the  methods 
of  the  ancients,  so  that  in  his  preface  to  his  "Mechanique 
Analytique,"  published  in  1788,  Lagrange  boasts,  "  One 
will  find  no  figures  in  this  work."  But  at  the  close  of 
the  eighteenth  century  the  field  opened  up  to  research 
by  the  invention  of  the  calculus  began  to  appear  so 
thoroughly  explored  that  new  methods  and  new  objects 


112  PROJECTIVE  GEOMETRY 

of  investigation  began  to  attract  attention.  Lagrange 
himself,  in  his  later  years,  turned  in  weariness  from 
analysis  and  mechanics,  and  applied  himself  to  chemistry, 
physics,  and  philosophical  speculations.  "  This  state  of 
mind,"  says  Darboux,*  "  we  find  almost  always  at  certain 
moments  in  the  lives  of  the  greatest  scholars."  At  any 
rate,  after  lying  fallow  for  almost  two  centuries,  the 
field  of  pure  geometry  was  attacked  with  almost  religious 
enthusiasm. 

177.  Newton  and  Maclaurin.  But  in  hastening  on 
to  the  epoch  of  Poncelet  and  Steiner  we  should  not 
omit  to  mention  the  work  of  Xewton  and  Maclaurin. 
Although  their  results  were  obtained  by  analysis  for  the 
most  part,  nevertheless  they  have  given  us  theorems 
which  fall  naturally  into  the  domain  of  synthetic  pro- 
jective geometry.  Thus  Newton's  "  organic  method "  f 
of  generating  conic  sections  is  closely  related  to  the 
method  which  we  have  made  use  of  in  Chapter  III. 
It  is  as  follows:  If  two  angles,  AOS  and  AO'S,  of  given 
magnitudes  turn  about  their  respective  vertices,  0  and  O', 
in  such  a  wag  that  the  point  of  intersection,  S,  of  one  pair 
of  arms  alivags  lies  on  a  straight  line,  the  point  of  inter- 
section, A,  of  the  other  pair  of  arms  will  describe  a  conic. 
The  proof  of  this  is  left  to  the  student. 

178.  Another  method  of  generating  a  conic  is  due  to 

Maclaurin. :f    The  construction,  which  we  also  leave  for 

the  student  to  justify,  is  as  follows :  If  a  triangle  C'PQ 

move  in  such  a  wag  that  its  sides,  PQ,  QC',  and  C'P,  turn 

*  See  Ball,  History  of  Mathematics,  French  edition,  t.  II,  233. 
f  Newton,  Principia,  lib.  i,  lemma  XXI. 

J  Maclaurin,  Philosophical  Transactions  of  the  Royal  Society  of 
London,  1735. 


SYNTHETIC  PROJECTIVE  GEOMETRY      113 

around  three  fixed  points,  R,  A,  B,  respectively,  while  two  of 
its  vertices,  P,  Q,  slide  along  two  fixed  lines,  CB'  and  CA', 
respectively,  then  the  remaining  vertex  will  describe  a  conic. 
179.  Descriptive  geometry  and  the  second  revival. 
The  second  revival  of  pure  geometry  was  again  to  take 
place  at  a  time  of  great  intellectual  activity.  The  period 
at  the  close  of  the  eighteenth  and  the  beginning  of 
the  nineteenth  century  is  adorned  with  a  glorious  list 
of  mighty  names,  among  which  are  Gauss,  Lagrange, 
Legendre,  Laplace,  Monge,  Carnot,  Poncelet,  Cauchy, 
Fourier,  Steiner,  Von  Staudt,  Mobius,  Abel,  and  many 
others.  The  renaissance  may  be  said  to  date  from  the  in- 
vention by  Monge  *  of  the  theory  of  descriptive  geometry. 
Descriptive  geometry  is  concerned  with  the  representa- 
tion of  figures  in  space  of  three  dimensions  by  means 
of  space  of  two  dimensions.  The  method  commonly 
used  consists  in  projecting  the  space  figure  on  two 
planes  (a  vertical  and  a  horizontal  plane  being  most 
convenient),  the  projections  being  made  most  simply 
for  metrical  purposes  from  infinity  in  directions  perpen- 
dicular to  the  two  planes  of  projection.  These  two 
planes  are  then  made  to  coincide  by  revolving  the  hori- 
zontal into  the  vertical  about  their  common  line.  Such 
is  the  method  of  descriptive  geometry  which  in  the 
hands  of  Monge  acquired  wonderful  generality  and  ele- 
gance. Problems  concerning  fortifications  were  worked 
so  quickly  by  this  method  that  the  commandant  at  the 
military  school  at  Mezieres,  where  Monge  was  a  drafts- 
man and  pupil,  viewed  the  results  with  distrust.  Monge 
afterward  became  professor  of  mathematics  at  Mezieres 
*  Monge,  G6ou\6tvie  Descriptive.    1800. 


114  PROJECTIVE  GEOMETRY 

and  gathered  around  him  a  group  of  students  destined 
to  have  a  share  in  the  advancement  of  pure  geometry. 
Among  these  were  Hachette,  Brianchon,  Dupin,  Chasles, 
Poncelet,  and  many  others. 

180.  Duality,  homology,  continuity,  contingent  rela- 
tions. Analytic  geometry  had  left  little  to  do  in  the 
way  of  discovery  of  new  material,  and  the  mathemati- 
cal world  was  ready  for  the  construction  of  the  edifice. 
The  activities  of  the  group  of  men  that  followed  Monge 
were  directed  toward  this  end,  and  we  now  begin  to 
hear  of  the  great  unifying  notions  of  duality,  homol- 
ogy, continuity,  contingent  relations,  and  the  like.  The 
devotees  of  pure  geometry  were  beginning  to  feel  the 
need  of  a  basis  for  their  science  which  should  be  at 
once  as  general  and  as  rigorous  as  that  of  the  analysts. 
Their  dream  was  the  building  up  of  a  system  of  geom- 
etry which  should  be  independent  of  analysis.  Monge, 
and  after  him  Poncelet,  spent  much  thought  on  the  so- 
called  "principle  of  continuity,"  afterwards  discussed 
by  Chasles  under  the  name  of  the  "principle  of  con- 
tingent relations."  To  get  a  clear  idea  of  this  principle, 
consider  a  theorem  in  geometry  in  the  proof  of  which 
certain  auxiliary  elements  are  employed.  These  ele- 
ments do  not  appear  in  the  statement  of  the  theorem, 
and  the  theorem  might  possibly  be  proved  without  them. 
In  drawing  the  figure  for  the  proof  of  the  theorem, 
however,  some  of  these  elements  may  not  appear,  or, 
as  the  analyst  would  say,  they  become  imaginary.  "  No 
matter,"  says  the  principle  of  contingent  relations,  "the 
theorem  is  true,  and  the  proof  is  valid  whether  the 
elements  used  hi  the  proof  are  real  or  imaginary." 


SYNTHETIC  PROJECTIVE  GEOMETRY     115 

181.  Poncelet  and  Cauchy.  The  efforts  of  Poncelet 
to  compel  the  acceptance  of  this  principle  independent 
of  analysis  resulted  in  a  bitter  and  perhaps  fruitless 
controversy  between  him  and  the  great  analyst  Cauchy. 
In  his  review  of  Poncelet's  great  work  on  the  projec- 
tive properties  of  figures  *  Cauchy  says,  "  In  his  pre- 
liminary discourse  the  author  insists  once  more  on  the 
necessity  of  admitting  into  geometry  what  he  calls  the 
'  principle  of  continuity.'  We  have  already  discussed 
that  principle  .  .  .  and  we  have  found  that  that  prin- 
ciple is,  properly  speaking,  only  a  strong  induction, 
which  cannot  be  indiscriminately  applied  to  all  sorts  of 
questions  in  geometry,  nor  even  in  analysis.  The  rea- 
sons which  we  have  given  as  the  basis  of  our  opinion 
are  not  affected  by  the  considerations  which  the  author 
has  developed  in  his  Traite  des  Proprietes  Projectives 
des  Figures."  Although  this  principle  is  constantly  made 
use  of  at  the  present  day  in  all  sorts  of  investigations, 
careful  geometricians  are  in  agreement  with  Cauchy 
in  this  matter,  and  use  it  only  as  a  convenient  work- 
ing tool  for  purposes  of  exploration.  The  one-to-one 
correspondence  between  geometric  forms  and  algebraic 
analysis  is  subject  to  many  and  important  exceptions. 
The  field  of  analysis  is  much  more  general  than  the 
field  of  geometry,  and  while  there  may  be  a  clear 
notion  in  analysis  to  correspond  to  every  notion  in 
geometry,  the  opposite  is  not  true.  Thus,  in  analysis 
we  can  deal  with  four  coordinates  as  well  as  with 
three,  but  the  existence  of  a  space  of  four  dimensions 

*  Poncelet,  Traite"  des  Propri6te\s  Projectives  des  Figures.  1822. 
(See  p.  367,  Vol.  II,  of  the  edition  of  1866.) 


116  PROJECTIVE  GEOMETRY 

to  correspond  to  it  does  not  therefore  follow.  When 
the  geometer  speaks  of  the  two  real  or  imaginary  inter- 
sections of  a  straight  line  with  a  conic,  he  is  really 
speaking  the  language  of  algebra.  Apart  from  the 
algebra  involved,  it  is  the  height  of  absurdity  to  try  to 
distinguish  between  the  two  points  in  which  a  line 
fails  to  meet  a  conic! 

182.  The  work  of  Poncelet.  But  Poncelet's  right  to 
the  title  "The  Father  of  Modern  Geometry"  does  not 
stand  or  fall  with  the  principle  of  contingent  relations. 
In  spite  of  the  fact  that  he  considered  this  principle 
the  most  important  of  all  his  discoveries,  his  reputation 
rests  on  more  solid  foundations.  He  was  the  first  to 
study  figures  in  homology,  which  is,  in  effect,  the  colline- 
ation  described  in  §  175,  where  corresponding  points 
lie  on  straight  lines  through  a  fixed  point.  He  was  the 
first  to  give,  by  means  of  the  theory  of  poles  and  polars, 
a  transformation  by  which  an  element  is  transformed 
into  another  of  a  different  sort.  Point-to-point  trans- 
formations will  sometimes  generalize  a  theorem,  but 
the  transformation  discovered  by  Poncelet  may  throw  a 
theorem  into  one  of  an  entirely  different  aspect.  The 
principle  of  duality,  first  stated  in  definite  form  by 
Gergonne,*  the  editor  of  the  mathematical  journal  in 
which  Poncelet  published  his  researches,  was  based  by 
Poncelet  on  his  theory  of  poles  and  polars.  He  also  put 
into  definite  form  the  notions  of  the  infinitely  distant 
elements  in  space  as  all  lying  on  a  plane  at  infinity. 

183.  The  debt  which  analytic  geometry  owes  to  syn- 
thetic  geometry.    The    reaction    of   pure   geometry  on 

*  Gergonne,  Annates  de  Mathtmatiques,  XVI,  200.   1820. 


SYNTHETIC  PROJECTIVE  GEOMETRY      117 

analytic  geometry  is  clearly  seen  in  the  development  of 
the  notion  of  the  class  of  a  curve,  which  is  the  number 
of  tangents  that  may  be  drawn  from  a  point  in  a  plane 
to  a  given  curve  lying  in  that  plane.  If  a  point  moves 
along  a  conic,  it  is  easy  to  show  —  and  the  student 
is  recommended  to  furnish  the  .proof  —  that  the  polar 
line  with  respect  to  a  conic  remains  tangent  to  another 
conic.  This  may  be  expressed  by  the  statement  that  the 
conic  is  of  the  second  order  and  also  of  the  second  class. 
It  might  be  thought  that  if  a  point  moved  along  a 
cubic  curve,  its  polar  line  with  respect  to  a  conic  would 
remain  tangent  to  another  cubic  curve.  This  is  not  the 
case,  however,  and  the  investigations  of  Poncelet  and 
others  to  determine  the  class  of  a  given  curve  were 
afterward  completed  by  Plucker.  The  notion  of  geo- 
metrical transformation  led  also  to  the  very  important 
developments  in  the  theory  of  invariants,  which,  geo- 
metrically, are  the  elements  and  configurations  which 
are  not  affected  by  the  transformation.  The  anharmonic 
ratio  of  four  points  is  such  an  invariant,  since  it  remains 
unaltered  under  all  projective  transformations. 

184.  Steiner  and  his  work.  In  the  work  of  Poncelet 
and  his  contemporaries,  Chasles,  Brianchon,  Hachette, 
Dupin,  Gergonne,  and  others,  the  anharmonic  ratio  en- 
joyed a  fundamental  role.  It  is  made  also  the  basis  of 
the  great  work  of  Steiner,*  who  was  the  first  to  treat 
of  the  conic,  not  as  the  projection  of  a  circle,  but  as  the 
locus  of  intersection  of  corresponding  rays  of  two  pro- 
jective pencils.    Steiner  not  only  related  to  each  other, 

*  Steiner,  Systematische  Entwickelung  der  Abhangigkeit  geome- 
trisolier  Gestalten  von  einander.    1832. 


118  PROJECTIVE  GEOMETRY 

in  one-to-one  correspondence,  point-rows  and  pencils 
and  all  the  other  fundamental  forms,  but  he  set  into 
correspondence  even  corves  and  surfaces  of  higher  de- 
grees. This  new  and  fertile  conception  gave  him  an 
easy  and  direct  route  into  the  most  abstract  and  diffi- 
cult regions  of  pure  geometry.  Much  of  his  work  was 
given  without  any  indication  of  the  methods  by  which 
he  had  arrived  at  it,  and  many  of  his  results  have  only 
recently  been  verified. 

185.  Von  Staudt  and  his  work.  To  complete  the  the- 
ory of  geometry  as  we  have  it  to-day  it  only  remained 
to  free  it  from  its  dependence  on  the  semimetrical  basis 
of  the  enharmonic  ratio.  This  work  was  accomplished  by 
Von  Staudt,*  who  applied  himself  to  the  restatement 
of  the  theory  of  geometry  in  a  form  independent  of 
analytic  and  metrical  notions.  The  method  which  has 
been  used  in  Chapter  II  to  develop  the  notion  of  four 
harmonic  points  by  means  of  the  complete  quadrilateral 
is  due  to  Von  Staudt.  His  work  is  characterized  by  a 
most  remarkable  generality,  in  that  he  is  able  to  discuss 
real  and  imaginary  forms  with  equal  ease.  Thus  he 
assumes  a  one-to-one  correspondence  between  the  points 
and  lines  of  a  plane,  and  defines  a  conic  as  the  locus 
of  points  which  lie  on  their  corresponding  lines,  and  a 
pencil  of  rays  of  the  second  order  as  the  system  of  lines 
which  pass  through  their  corresponding  points.  The 
point-row  and  pencil  of  the  second  order  may  be  real 
or  imaginary,  but  his  theorems  still  apply.  An  illustra- 
tion of  a  correspondence  of  this  sort,  where  the  conic 
is  imaginary,  is  given  in  §  15  of  the  first  chapter.  In 
*  Von  Staudt,  Geometric  der  Lage.    1847. 


SYNTHETIC  PEOJECTIVE  GEOMETRY      119 

defining  conjugate  imaginary  points  on  a  line,  Von 
Staudt  made  use  of  an  involution  of  points  having  no 
double  points.  His  methods,  while  elegant  and  power- 
ful, are  hardly  adapted  to  an  elementary  course,  but 
Reye*  and  others  have  done  much  toward  simplifying 
his  presentation. 

186.  Recent  developments.  It  would  be  only  confus- 
ing to  the  student  to  attempt  to  trace  here  the  later 
developments  of  the  science  of  projective  geometry.  It 
is  concerned  for  the  most  part  with  curves  and  surfaces 
of  a  higher  degree  than  the  second.  Purely  synthetic 
methods  have  been  used  with  marked  success  in  the 
study  of  the  straight  line  in  space.  The  struggle  be- 
tween analysis  and  pure  geometry  has  long  since  come 
to  an  end.  Each  has  its  distinct  advantages,  and  the 
mathematician  who  cultivates  one  at  the  expense  of  the 
other  will  never  attain  the  results  that  he  would  attain 
if  both  methods  were  equally  ready  to  his  hand.  Pure 
geometry  has  to  its  credit  some  of  the  finest  discov- 
eries in  mathematics,  and  need  not  apologize  for  having 
been  born.  The  day  of  its  usefulness  has  not  passed 
with  the  invention  of  abridged  notation  and  of  short 
methods  in  analysis.  While  we  may  be  certain  that  any 
geometrical  problem  may  always  be  stated  in  analytic 
form,  it  does  not  follow  that  that  statement  will  be 
simple  or  easily  interpreted.  For  many  mathematicians 
the  geometric  intuitions  are  weak,  and  for  such  the 
method  will  have  little  attraction.  On  the  other  hand, 
there  will  always  be  those  for  whom  the  subject  will 
have  a  peculiar  glamor  —  who  will  follow  with  delight 
*  Keye,  Geometrie  der  Lage.   Translated  by  Holgate,  1897. 


120  PROJECTIVE  GEOMETRY 

the  curious  and  unexpected  relations  between  the  forms 
of  space.  There  is  a  corresponding  pleasure,  doubtless, 
for  the  analyst  in  tracing  the  marvelous  connections 
between  the  various  fields  in  which  he  wanders,  and  it 
is  as  absurd  to  shut  one's  eyes  to  the  beauties  in  one 
as  it  is  to  ignore  those  in  the  other.  "Let  us  cultivate 
geometry,  then,"  says  Darboux,*  "without  wishing  in 
all  points  to  equal  it  to  its  rival.  Besides,  if  we  were 
tempted  to  neglect  it,  it  would  not  be  long  in  finding 
in  the  applications  of  mathematics,  as  once  it  has  al- 
ready done,  the  means  of  renewing  its  life  and  of 
developing  itself  anew.  It  is  like  the  Giant  Antaeus, 
who  renewed  his  strength  by  touching  the  earth." 

*  Ball,  loc.  cit.  p.  261. 


INDEX 


(The  numbers  refer  to  the  paragraphs) 


Abel  (1802-1829),  179 

Analogy,  24 

Analytic  geometry,  21,  118,  119, 

120,  146,  176,  180 
Anharmonic  ratio,46,161, 184,185 
Apollonius  (second  half  of  third 

century  b.  c),  70 
Archimedes  (287-212  B.C.),  176 
Aristotle  (384-322  B.C.),  169 
Asymptotes,  111,  113,  114,  115, 

116,  117,  118,  148 
Axes  of  a  conic,  148 
Axial  pencil,  7,  8,  23,  50,  54 
Axis  of  perspectivity,  8,  47 

Bacon  (1561-1626),  162 
Bisection,  41,  109 
Brianchon  (1785-1864),  84,  85,  86, 
88,  89,  90,  95,  105,  113,  174,  184 

Calculus,  176 

Carnot  (1796-1832),  179 

Cauchy  (1789-1857),  179, 181 

Cavalieri  (1598-1647),  162 

Center  of  a  conic,  107,  112,  148 

Center  of  involution,  141,  142 

Center  of  perspectivity,  8 

Central  conic,  120 

Chasles  (1793-1880),  168, 179, 180, 

184 
Circle,  21,  73,  80,  145,  146,  147 
Circular  involution,  147, 149,  150, 

151 
Circular  points,  146 
Class  of  a  curve,  183 
Classification  of  conies,  110 
Collineation,  175 
Concentric  pencils,  50 
Cone  of  the  second  order,  59 


Conic,  73,  81 

Conjugate  diameters,  114, 148 
Conjugate  normal,  151 
Conjugate  points  and  lines,  100, 

109,  138,  139,  140 
Constants  in  an  equation,  21 
Contingent  relations,  180,  181 
Continuity,  180,  181 
Continuous  correspondence,  9, 10, 

21,49 
Corresponding  elements,  64 
Counting,  1,  4 
Cross  ratio,  46 

Darboux,  176, 186 

De  Beaugrand,  170 

Degenerate  pencil  of  rays  of  the 

second  order,  58,  93 
Degenerate    point-row    of    the 

second  order,  56,  78 
De  la  Hire  (1640-1718),  168, 171, 

175 
Desanrues  (1593-1662),  25,  26,  40, 

121,"l25, 162,  163, 164, 165,  166, 

167, 168,  169,  170, 171, 174,  175 
Descartes  (1596-1650),  162,  170, 

171,  174,  176 
Descriptive  geometry,  179 
Diameter,  107 
Directrix,  157,  158,  159,  160 
Double  correspondence,  128,  130 
Double  points  of  an  involution,  124 
Double  rays  of  an  involution,  133, 

134 
Duality,  94,  104,  161,  180, 182 
Dupin  (1784-1873),  174,  184 

Eccentricity  of  conic,  159 
Ellipse,  110,  111,  162 


121 


122 


PROJECTIVE  GEOMETRY 


Equation  of  conic,  118,  119,  120 
Euclid  (ca.  300  b.c),  6,  22,  104 
Euler  (1707-1783),  166 

Eermat  (1601-1665),  162,  171 
Foci  of  a  conic,  152, 153, 154, 155, 

156,  157,  158,  159,  160,  161,  162 
Fourier  (1768-1830),  179 
Fourth  harmonic,  29 
Fundamental  form,  7, 16,  23,  36, 

47,  60, 184 

Galileo  (1564-1642),  162, 169, 170, 

176 
Gauss  (1777-1855),  179 
Gergonne  (1771-1859),  182,  184 
Greek  geometry,  161 

Hachette  (1769-1834),  179,  184 
Harmonic  conjugates,  29,  30,  39 
Harmonic  elements,  36,  49,  91, 

163,  185 
Harmonic  lines,  33,  34,  35,  66,  67 
Harmonic  planes,  34,  35 
Harmonic  points,  29,  31,  32,  33, 

34,  35,  36,  43,  71,  161 
Harmonic  tangents  to  a  conic, 

91,92 
Harvey  (1578-1657),  169 
Homology,  180,  182 
Huygena  (1629-1695),  162 
Hyperbola,  110,111,113, 114, 115, 

116,  117,  118,  162 

Imaginary  elements,  146, 180, 181, 

182, 185 
Infinitely  distant  elements,  6,  9, 

22,  39,  40,  41,  104,  107,  110 
Infinity,  4,  5,  10,  12,  13,  14,  15, 

17,  18,  19,  20,  21,  22,  41 
Involution,  37,  123,  124,  125,  126. 

127, 128,  129,  130, 131, 132,  133, 

134,  135, 136,  137, 138, 139, 140, 

101,  163,  170 

Kepler  (1571-1630),  162 

Lagrange  (1736-1813),  176, 179 
Laplace  (1749-1827),  179 
Legendre  (1752-1833),  179 


Leibniz  (1646-1716),  173 

Linear  construction,  40,  41,  42 

Maclaurin  (1698-1746),  177,  178 
Measurements,  23,  40,  41, 104 
Mersenne  (1588-1648).  I»i8.  171 
Metrical  theorems,  40,  104,  106, 

107,  141 
Middle  point,  39,  41 
Mobius  (1790-1868),  17!) 
Monge  (1746-1818),  179,  180 

Napier  (1550-1617),  162 
Newton  (1642-1727),  177 
Numbers,  4,  21,  43 
Numerical  computations,  43,  44, 
46 

One-to-one  correspondence,  I.  2, 

3,  4,  •-..  6,  7.  :».  10,  11.  24,  86, 

37,  43,  60,  104,  106,  184 
Opposite  sides  of  a  hexagon,  70 
Opposite  sides  of  a  quadrilateral, 

28,  29 
Order  of  a  form,  7,  10,  11,  12,  13, 

14,  15,  16,  17,  18, 19,  20,  21 

Pappus    (fourth    century    a.i».). 

161 
Parabola,  110,  111,  112,  119,  162 
Parallel  lines,  39,  41,  162 
Pascal  (1623-1662),  69,  70,  74,  75, 

76,   77,   78,  95,  105,   125,   162, 

169,  171,  172,  173 
Pencil  of  planes  of   the  second 

order,  59 
Pencil  of  rays,  6,  7,  8,  23  ;  of  the 

second  order,  57,  60,  79,  81 
Perspective  position,  6,  8,  35,  37, 

51,  53,  71 
Plane  system,  16,  23 
Planes  on  space,  17 
Point  of  contact,  87,  88,  89,  90 
Point  system,  16,  23 
Point^row,  6,  7,  8,  9,  23  ;  of  the 

second  order,  55,  60,  61,  66, 

67,  72 
Points  in  space,  18 
Pole  and  polar,  98,  99,  100,  101, 

138,  104,  166 


IXDEX 


123 


Poncelet  (1788-1807),    177,    17!), 

180,  181,  182,  183,  184 
Principal  axis  of  a  conic,  157 
Projection,  161 

Projective  axial  pencils,  59 
Projective  correspondence,  9,  35, 

36,  37,  47,  71,  92,  104 
Projective  pencils,  53,  64,  68 
Projective  point-rows,  51,  79 
Projective  properties,  24 
Projective  theorems,  40,  104 

Quadrangle,  26,  27,  28,  29 
Quadric  cone,  59 
Quadrilateral,  88,  95,  <)G 

Koberval  (1602-1675),  168 
Ruler  construction,  40 

Scheiner,  169 

Self-corresponding  elements,  47, 
48,  49,  50,  51 

Self-dual,  105 

Self-polar  triangle,  102 

Separation  of  elements  in  involu- 
tion, 148 


Separation  of  harmonic  conju- 
gates, 38         o 

Sequence  of  points,  49 

Sign  of  segment,  44,  45 

Similarity,  106 

Skew  lines,  12 

Space  system,  19,  23 

Sphere,  21 

Steiner (1796-1863),  129, 130, 131, 
177,  179,  184 

Steiner's  construction,  129,  130, 
131 

Superposed  point- rows,  47,  48,  49 

Surfaces  of  the  second  degree,  166 

System  of  lines  in  space,  20,  23 

Systems  of  conies,  125 

Tangent  line,  61,  80,  81,  87,  88, 

89,  90,  91,  92 
Tycho  Brahe  (1546-1601),  162 

Verner,  161 

Vertex  of  conic,  157,  159 

Von  Staudt  (1798-1867),  179,  185 

Wallis  (1616-1703),  162 


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